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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Gelander%2C+T&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Gelander%2C+T&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Gelander%2C+T&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.07033">arXiv:2411.07033</a> <span> [<a href="https://arxiv.org/pdf/2411.07033">pdf</a>, <a href="https://arxiv.org/ps/2411.07033">ps</a>, <a href="https://arxiv.org/format/2411.07033">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Spectral gap for products and a strong normal subgroup theorem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bader%2C+U">Uri Bader</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Levit%2C+A">Arie Levit</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.07033v2-abstract-short" style="display: inline;"> We establish a general spectral gap theorem for actions of products of groups which may replace Kazhdan's property (T) in various situations. As a main application, we prove that a confined subgroup of an irreducible lattice in a higher rank semisimple Lie group is of finite index. This significantly strengthens the classical normal subgroup theorem of Margulis and removes the property (T) assumpt… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.07033v2-abstract-full').style.display = 'inline'; document.getElementById('2411.07033v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.07033v2-abstract-full" style="display: none;"> We establish a general spectral gap theorem for actions of products of groups which may replace Kazhdan's property (T) in various situations. As a main application, we prove that a confined subgroup of an irreducible lattice in a higher rank semisimple Lie group is of finite index. This significantly strengthens the classical normal subgroup theorem of Margulis and removes the property (T) assumption from the recent counterpart result of Fraczyk and Gelander. We further show that any confined discrete subgroup of a higher rank semisimple Lie group satisfying a certain irreducibility condition is an irreducible lattice. This implies a variant of the Stuck-Zimmer conjecture under a strong irreducibility assumption of the action. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.07033v2-abstract-full').style.display = 'none'; document.getElementById('2411.07033v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">very minor changes from previous version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G25 22E40 37A05 37B99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.21208">arXiv:2407.21208</a> <span> [<a href="https://arxiv.org/pdf/2407.21208">pdf</a>, <a href="https://arxiv.org/format/2407.21208">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Things we can learn by considering random locally symmetric manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.21208v2-abstract-short" style="display: inline;"> In recent years various results about locally symmetric manifolds were proven using probabilistic approaches. One of the approaches is to consider random manifolds by associating a probability measure to the space of discrete subgroups of the isometry Lie group. The main goals are to prove results about deterministic groups and manifolds by considering appropriate measures. In this overview paper… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.21208v2-abstract-full').style.display = 'inline'; document.getElementById('2407.21208v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.21208v2-abstract-full" style="display: none;"> In recent years various results about locally symmetric manifolds were proven using probabilistic approaches. One of the approaches is to consider random manifolds by associating a probability measure to the space of discrete subgroups of the isometry Lie group. The main goals are to prove results about deterministic groups and manifolds by considering appropriate measures. In this overview paper we describe several such results, observing the evolution process of the measures involved. Starting with a result whose proof considered finitely supported measures (more precisely, measures supported on finitely many conjugacy classes) and proceeding with results which were outcome of the successful and popular theory of IRS (invariant random subgroups). In the last couple of years the theory has expanded to SRS (stationary random subgroups) allowing to deal with a lot more problems and establish stronger results. In the last section we shall review a very recent (yet unpublished) result whose proof make use of random subgroups which are not even stationary. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.21208v2-abstract-full').style.display = 'none'; document.getElementById('2407.21208v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">45 pages, several figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> proceedings of the ICTS, volume on Zariski dense subgroups, number theory and geometric, 2024 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.15867">arXiv:2402.15867</a> <span> [<a href="https://arxiv.org/pdf/2402.15867">pdf</a>, <a href="https://arxiv.org/ps/2402.15867">ps</a>, <a href="https://arxiv.org/format/2402.15867">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> An Invitation to Analytic Group Theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cohen%2C+T">Tal Cohen</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.15867v1-abstract-short" style="display: inline;"> This book is concerned with analytic approaches of studying groups and their actions. Much attention is devoted to the study of amenability and Kazhdan's property (T), which are perhaps the most important analytic properties of a group, but we also discuss other analytic notions. We tried to introduce tricks, ideas and lemmas that repeatedly turn out to be useful in various situations. Our main gu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.15867v1-abstract-full').style.display = 'inline'; document.getElementById('2402.15867v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.15867v1-abstract-full" style="display: none;"> This book is concerned with analytic approaches of studying groups and their actions. Much attention is devoted to the study of amenability and Kazhdan's property (T), which are perhaps the most important analytic properties of a group, but we also discuss other analytic notions. We tried to introduce tricks, ideas and lemmas that repeatedly turn out to be useful in various situations. Our main guideline was to expose the beauty of the theory and to present many different aspects of it while keeping the text short, simple and accessible, sometimes at the expense of diving deep or providing thorough expositions. Hopefully this book could serve as a smooth entry to Analytic Group Theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.15867v1-abstract-full').style.display = 'none'; document.getElementById('2402.15867v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.15976">arXiv:2311.15976</a> <span> [<a href="https://arxiv.org/pdf/2311.15976">pdf</a>, <a href="https://arxiv.org/ps/2311.15976">ps</a>, <a href="https://arxiv.org/format/2311.15976">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> A Quantitative Selberg's Lemma </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Slutsky%2C+R">Raz Slutsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.15976v2-abstract-short" style="display: inline;"> We show that an arithmetic lattice $螕$ in a semi-simple Lie group $G$ contains a torsion-free subgroup of index $未(v)$ where $v = 渭(G/螕)$ is the co-volume of the lattice. We prove that $未$ is polynomial in general and poly-logarithmic under GRH. We then show that this poly-logarithmic bound is almost optimal, by constructing certain lattices with torsion elements of order… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.15976v2-abstract-full').style.display = 'inline'; document.getElementById('2311.15976v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.15976v2-abstract-full" style="display: none;"> We show that an arithmetic lattice $螕$ in a semi-simple Lie group $G$ contains a torsion-free subgroup of index $未(v)$ where $v = 渭(G/螕)$ is the co-volume of the lattice. We prove that $未$ is polynomial in general and poly-logarithmic under GRH. We then show that this poly-logarithmic bound is almost optimal, by constructing certain lattices with torsion elements of order $\sim \frac{\log v}{\log \log v}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.15976v2-abstract-full').style.display = 'none'; document.getElementById('2311.15976v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40; 20G30; 20H05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.11857">arXiv:2212.11857</a> <span> [<a href="https://arxiv.org/pdf/2212.11857">pdf</a>, <a href="https://arxiv.org/ps/2212.11857">ps</a>, <a href="https://arxiv.org/format/2212.11857">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Aut(Fn) actions on representation spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.11857v2-abstract-short" style="display: inline;"> J. Wiegold conjectured that if n>2 and G is a finite simple group, then the action of Aut(F_n) on Epi(F_n,G) is transitive. In this note we consider analogous questions where G is a compact Lie group, a non-compact simple analytic group or a simple algebraic group. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.11857v2-abstract-full" style="display: none;"> J. Wiegold conjectured that if n>2 and G is a finite simple group, then the action of Aut(F_n) on Epi(F_n,G) is transitive. In this note we consider analogous questions where G is a compact Lie group, a non-compact simple analytic group or a simple algebraic group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.11857v2-abstract-full').style.display = 'none'; document.getElementById('2212.11857v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.10677">arXiv:2106.10677</a> <span> [<a href="https://arxiv.org/pdf/2106.10677">pdf</a>, <a href="https://arxiv.org/ps/2106.10677">ps</a>, <a href="https://arxiv.org/format/2106.10677">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Bounds on Systoles and Homotopy Complexity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Vollrath%2C+P">Paul Vollrath</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.10677v3-abstract-short" style="display: inline;"> We give an elementary proof of a weak variant of the homotopy type conjecture from [Gel04]. The proof relies on Belolipetskys estimate on the systole in terms of the volume which is a consequence of Prasads volume formula. In particular we extend the estimate established in [B20] for hyperbolic manifolds to general arithmetic manifolds. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.10677v3-abstract-full" style="display: none;"> We give an elementary proof of a weak variant of the homotopy type conjecture from [Gel04]. The proof relies on Belolipetskys estimate on the systole in terms of the volume which is a consequence of Prasads volume formula. In particular we extend the estimate established in [B20] for hyperbolic manifolds to general arithmetic manifolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.10677v3-abstract-full').style.display = 'none'; document.getElementById('2106.10677v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.11875">arXiv:2103.11875</a> <span> [<a href="https://arxiv.org/pdf/2103.11875">pdf</a>, <a href="https://arxiv.org/ps/2103.11875">ps</a>, <a href="https://arxiv.org/format/2103.11875">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Effective discreteness radius of stabilisers for stationary actions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Levit%2C+A">Arie Levit</a>, <a href="/search/math?searchtype=author&query=Margulis%2C+G">Gregory Margulis</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2103.11875v1-abstract-short" style="display: inline;"> We prove an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $尾r^未$ for some explicit constants $尾, 未> 0$ depending only the group. This is a consequence of a key convolution inequ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.11875v1-abstract-full').style.display = 'inline'; document.getElementById('2103.11875v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.11875v1-abstract-full" style="display: none;"> We prove an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $尾r^未$ for some explicit constants $尾, 未> 0$ depending only the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a novel proof of the fact that all lattices in semisimple groups are weakly cocompact. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.11875v1-abstract-full').style.display = 'none'; document.getElementById('2103.11875v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">43 pages. 1 appendix</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40; 22E30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.00640">arXiv:2101.00640</a> <span> [<a href="https://arxiv.org/pdf/2101.00640">pdf</a>, <a href="https://arxiv.org/ps/2101.00640">ps</a>, <a href="https://arxiv.org/format/2101.00640">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Infinite Volume and Infinite Injectivity Radius </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fraczyk%2C+M">Mikolaj Fraczyk</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.00640v3-abstract-short" style="display: inline;"> We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group and let $螞\le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $螞\backslash G/K$ admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not n… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.00640v3-abstract-full').style.display = 'inline'; document.getElementById('2101.00640v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.00640v3-abstract-full" style="display: none;"> We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group and let $螞\le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $螞\backslash G/K$ admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan's property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the St眉ck-Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.00640v3-abstract-full').style.display = 'none'; document.getElementById('2101.00640v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">We added some examples (in particular 6.8) to the previous version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40; 53C30; 53C35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.09361">arXiv:1909.09361</a> <span> [<a href="https://arxiv.org/pdf/1909.09361">pdf</a>, <a href="https://arxiv.org/ps/1909.09361">ps</a>, <a href="https://arxiv.org/format/1909.09361">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Maximal subgroups of countable groups, a survey </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Glasner%2C+Y">Yair Glasner</a>, <a href="/search/math?searchtype=author&query=Soifer%2C+G">Gregory Soifer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1909.09361v2-abstract-short" style="display: inline;"> This paper is a survey on the works [MS77, MS79, MS81] on maximal subgroups in finitely generated linear groups, and the works that followed it [GG08, GG13b, GG13a, Kap03, Iva92, HO16, GM16, AGS14, Sf90, Sf98, Per05, AKT16, FG18, GS17] concerning maximal subgroups of infinite index in linear groups as well as in various other groups possessing a suitable geometry or dynamics. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.09361v2-abstract-full" style="display: none;"> This paper is a survey on the works [MS77, MS79, MS81] on maximal subgroups in finitely generated linear groups, and the works that followed it [GG08, GG13b, GG13a, Kap03, Iva92, HO16, GM16, AGS14, Sf90, Sf98, Per05, AKT16, FG18, GS17] concerning maximal subgroups of infinite index in linear groups as well as in various other groups possessing a suitable geometry or dynamics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.09361v2-abstract-full').style.display = 'none'; document.getElementById('1909.09361v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">40 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 20B15; Secondary 20B10; 20B07; 20E28 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.04828">arXiv:1903.04828</a> <span> [<a href="https://arxiv.org/pdf/1903.04828">pdf</a>, <a href="https://arxiv.org/ps/1903.04828">ps</a>, <a href="https://arxiv.org/format/1903.04828">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> On the Minimal Size of a Generating Set of Lattices in Lie Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Slutsky%2C+R">Raz Slutsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1903.04828v2-abstract-short" style="display: inline;"> We prove that the rank (that is, the minimal size of a generating set) of lattices in a general connected Lie group is bounded by the co-volume of the projection of the lattice to the semi-simple part of the group. This was proved by Gelander for semi-simple Lie groups and by Mostow for solvable Lie groups. Here we consider the general case, relying on the semi-simple case. In particular, we exten… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.04828v2-abstract-full').style.display = 'inline'; document.getElementById('1903.04828v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.04828v2-abstract-full" style="display: none;"> We prove that the rank (that is, the minimal size of a generating set) of lattices in a general connected Lie group is bounded by the co-volume of the projection of the lattice to the semi-simple part of the group. This was proved by Gelander for semi-simple Lie groups and by Mostow for solvable Lie groups. Here we consider the general case, relying on the semi-simple case. In particular, we extend Mostow's theorem from solvable to amenable groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.04828v2-abstract-full').style.display = 'none'; document.getElementById('1903.04828v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 September, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Lie Theory, 30(1), 33-40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.02520">arXiv:1811.02520</a> <span> [<a href="https://arxiv.org/pdf/1811.02520">pdf</a>, <a href="https://arxiv.org/ps/1811.02520">ps</a>, <a href="https://arxiv.org/format/1811.02520">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Convergence of normalized Betti numbers in nonpositive curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Abert%2C+M">Miklos Abert</a>, <a href="/search/math?searchtype=author&query=Bergeron%2C+N">Nicolas Bergeron</a>, <a href="/search/math?searchtype=author&query=Biringer%2C+I">Ian Biringer</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.02520v3-abstract-short" style="display: inline;"> We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X \neq \mathbb H^3$, and $(M_n)$ is any Benjamini-Schramm convergent sequence of finite volume $X$-manifolds, then the normalized Betti numbers… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.02520v3-abstract-full').style.display = 'inline'; document.getElementById('1811.02520v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.02520v3-abstract-full" style="display: none;"> We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X \neq \mathbb H^3$, and $(M_n)$ is any Benjamini-Schramm convergent sequence of finite volume $X$-manifolds, then the normalized Betti numbers $b_k(M_n)/vol(M_n)$ converge for all $k$. As a corollary, if $X$ has higher rank and $(M_n)$ is any sequence of distinct, finite volume $X$-manifolds, the normalized Betti numbers of $M_n$ converge to the $L^2$ Betti numbers of $X$. This extends our earlier work with Nikolov, Raimbault and Samet, where we proved the same convergence result for uniformly thick sequences of compact $X$-manifolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.02520v3-abstract-full').style.display = 'none'; document.getElementById('1811.02520v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">54 pages, previous version fixed an error in Section 4.1 and corrected some typos. This version reworks Section 2, including details for the proof of continuity of our construction</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1807.06979">arXiv:1807.06979</a> <span> [<a href="https://arxiv.org/pdf/1807.06979">pdf</a>, <a href="https://arxiv.org/format/1807.06979">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> A view on Invariant Random Subgroups and Lattices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1807.06979v1-abstract-short" style="display: inline;"> For more than half a century lattices in Lie groups played an important role in geometry, number theory and group theory. Recently the notion of Invariant Random Subgroups (IRS) emerged as a natural generalization of lattices. It is thus intriguing to extend results from the theory of lattices to the context of IRS, and to study lattices by analyzing the compact space of all IRS of a given group.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.06979v1-abstract-full').style.display = 'inline'; document.getElementById('1807.06979v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1807.06979v1-abstract-full" style="display: none;"> For more than half a century lattices in Lie groups played an important role in geometry, number theory and group theory. Recently the notion of Invariant Random Subgroups (IRS) emerged as a natural generalization of lattices. It is thus intriguing to extend results from the theory of lattices to the context of IRS, and to study lattices by analyzing the compact space of all IRS of a given group. This article focuses on the interplay between lattices and IRS, mainly in the classical case of semisimple analytic groups over local fields. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.06979v1-abstract-full').style.display = 'none'; document.getElementById('1807.06979v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 July, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Submitted to the proceeding of the ICM, 2018. The talk in the congress will be on other stuff</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.05151">arXiv:1709.05151</a> <span> [<a href="https://arxiv.org/pdf/1709.05151">pdf</a>, <a href="https://arxiv.org/ps/1709.05151">ps</a>, <a href="https://arxiv.org/format/1709.05151">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Gasch眉tz Lemma for Compact Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cohen%2C+T">Tal Cohen</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.05151v2-abstract-short" style="display: inline;"> We prove the Gasch眉tz Lemma holds for all metrisable compact groups. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.05151v2-abstract-full" style="display: none;"> We prove the Gasch眉tz Lemma holds for all metrisable compact groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.05151v2-abstract-full').style.display = 'none'; document.getElementById('1709.05151v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Changed terminology and corrected typos</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1707.03578">arXiv:1707.03578</a> <span> [<a href="https://arxiv.org/pdf/1707.03578">pdf</a>, <a href="https://arxiv.org/ps/1707.03578">ps</a>, <a href="https://arxiv.org/format/1707.03578">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="K-Theory and Homology">math.KT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Invariant random subgroups over non-Archimedean local fields </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Levit%2C+A">Arie Levit</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1707.03578v2-abstract-short" style="display: inline;"> Let $G$ be a higher rank semisimple linear algebraic group over a non-Archimedean local field. The simplicial complexes corresponding to any sequence of pairwise non-conjugate irreducible lattices in $G$ are Benjamini-Schramm convergent to the Bruhat-Tits building. Convergence of the relative Plancherel measures and normalized Betti numbers follows. This extends the work of Abert, Bergeron, Biring… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.03578v2-abstract-full').style.display = 'inline'; document.getElementById('1707.03578v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1707.03578v2-abstract-full" style="display: none;"> Let $G$ be a higher rank semisimple linear algebraic group over a non-Archimedean local field. The simplicial complexes corresponding to any sequence of pairwise non-conjugate irreducible lattices in $G$ are Benjamini-Schramm convergent to the Bruhat-Tits building. Convergence of the relative Plancherel measures and normalized Betti numbers follows. This extends the work of Abert, Bergeron, Biringer, Gelander, Nokolov, Raimbault and Samet from real Lie groups to linear groups over arbitrary local fields. Along the way, various results concerning Invariant Random Subgroups and in particular a variant of the classical Borel density theorem are also extended. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.03578v2-abstract-full').style.display = 'none'; document.getElementById('1707.03578v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.00342">arXiv:1702.00342</a> <span> [<a href="https://arxiv.org/pdf/1702.00342">pdf</a>, <a href="https://arxiv.org/ps/1702.00342">ps</a>, <a href="https://arxiv.org/format/1702.00342">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> A note on local rigidity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bergeron%2C+N">Nicolas Bergeron</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1702.00342v1-abstract-short" style="display: inline;"> The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.00342v1-abstract-full" style="display: none;"> The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.00342v1-abstract-full').style.display = 'none'; document.getElementById('1702.00342v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 January, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This is a 2004 paper uploaded for archiving purposes</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Geom. Dedicata 107 (2004), 111-131 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.09510">arXiv:1612.09510</a> <span> [<a href="https://arxiv.org/pdf/1612.09510">pdf</a>, <a href="https://arxiv.org/ps/1612.09510">ps</a>, <a href="https://arxiv.org/format/1612.09510">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> On the growth of L2-invariants of locally symmetric spaces, II: exotic invariant random subgroups in rank one </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Abert%2C+M">Miklos Abert</a>, <a href="/search/math?searchtype=author&query=Bergeron%2C+N">Nicolas Bergeron</a>, <a href="/search/math?searchtype=author&query=Biringer%2C+I">Ian Biringer</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Nikolov%2C+N">Nikolay Nikolov</a>, <a href="/search/math?searchtype=author&query=Raimbault%2C+J">Jean Raimbault</a>, <a href="/search/math?searchtype=author&query=Samet%2C+I">Iddo Samet</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1612.09510v1-abstract-short" style="display: inline;"> In the first paper of this series (arxiv.org/abs/1210.2961) we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo-Stuck-Zimmer… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.09510v1-abstract-full').style.display = 'inline'; document.getElementById('1612.09510v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.09510v1-abstract-full" style="display: none;"> In the first paper of this series (arxiv.org/abs/1210.2961) we studied the asymptotic behavior of Betti numbers, twisted torsion and other spectral invariants for sequences of lattices in Lie groups G. A key element of our work was the study of invariant random subgroups (IRSs) of G. Any sequence of lattices has a subsequence converging to an IRS, and when G has higher rank, the Nevo-Stuck-Zimmer theorem classifies all IRSs of G. Using the classification, one can deduce asymptotic statments about spectral invariants of lattices. When G has real rank one, the space of IRSs is more complicated. We construct here several uncountable families of IRSs in the groups SO(n,1). We give dimension-specific constructions when n=2,3, and also describe a general gluing construction that works for every n at least 2. Part of the latter construction is inspired by Gromov and Piatetski-Shapiro's construction of non-arithmetic lattices in SO(n,1). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.09510v1-abstract-full').style.display = 'none'; document.getElementById('1612.09510v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages. An earlier version of this appeared as sections 11--13 of arXiv:1210.2961</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40; 57M50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.06220">arXiv:1612.06220</a> <span> [<a href="https://arxiv.org/pdf/1612.06220">pdf</a>, <a href="https://arxiv.org/ps/1612.06220">ps</a>, <a href="https://arxiv.org/format/1612.06220">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Lattices in amenable groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bader%2C+U">U. Bader</a>, <a href="/search/math?searchtype=author&query=Caprace%2C+P">P-E. Caprace</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">T. Gelander</a>, <a href="/search/math?searchtype=author&query=Mozes%2C+S">Sh. Mozes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1612.06220v3-abstract-short" style="display: inline;"> Let $G$ be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a non-Archimedean extension of Mostow's theorem by showing the amenable linear locally compact groups have property (M). However property (M) does not hold for… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.06220v3-abstract-full').style.display = 'inline'; document.getElementById('1612.06220v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.06220v3-abstract-full" style="display: none;"> Let $G$ be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a non-Archimedean extension of Mostow's theorem by showing the amenable linear locally compact groups have property (M). However property (M) does not hold for all solvable locally compact groups: indeed, we exhibit an example of a metabelian locally compact group with a non-uniform lattice. We show that compactly generated metabelian groups, and more generally nilpotent-by-nilpotent groups, do have property (M). Finally, we highlight a connection of property (M) with the subtle relation between the analytic notions of strong ergodicity and the spectral gap. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.06220v3-abstract-full').style.display = 'none'; document.getElementById('1612.06220v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">38 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22F30 (primary) 22D40; 43A07 (secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1612.04871">arXiv:1612.04871</a> <span> [<a href="https://arxiv.org/pdf/1612.04871">pdf</a>, <a href="https://arxiv.org/ps/1612.04871">ps</a>, <a href="https://arxiv.org/format/1612.04871">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> </div> <p class="title is-5 mathjax"> Homology and homotopy complexity in negative curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bader%2C+U">Uri Bader</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Sauer%2C+R">Roman Sauer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1612.04871v5-abstract-short" style="display: inline;"> Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, there is a sequence of $3$-dimensional hyperbolic manifolds that converges to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.04871v5-abstract-full').style.display = 'inline'; document.getElementById('1612.04871v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1612.04871v5-abstract-full" style="display: none;"> Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, there is a sequence of $3$-dimensional hyperbolic manifolds that converges to $\mathbb{H}^3$ in the Benjamini--Schramm topology while its normalized torsion in the first homology is dense in $[0,\infty]$. In dimension $d\geq 4$ a somewhat precise estimate is given for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension $d\ge 5$ up to homeomorphism. These results are based on an effective simplicial thick-thin decomposition which is of independent interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1612.04871v5-abstract-full').style.display = 'none'; document.getElementById('1612.04871v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">final version; to appear in JEMS</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 55N99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1611.08264">arXiv:1611.08264</a> <span> [<a href="https://arxiv.org/pdf/1611.08264">pdf</a>, <a href="https://arxiv.org/ps/1611.08264">ps</a>, <a href="https://arxiv.org/format/1611.08264">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Invariable generation of Thompson groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Golan%2C+G">Gili Golan</a>, <a href="/search/math?searchtype=author&query=Juschenko%2C+K">Kate Juschenko</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1611.08264v2-abstract-short" style="display: inline;"> A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this paper, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, where… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.08264v2-abstract-full').style.display = 'inline'; document.getElementById('1611.08264v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1611.08264v2-abstract-full" style="display: none;"> A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this paper, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.08264v2-abstract-full').style.display = 'none'; document.getElementById('1611.08264v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 November, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 November, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, v2: Typos corrected</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.01693">arXiv:1605.01693</a> <span> [<a href="https://arxiv.org/pdf/1605.01693">pdf</a>, <a href="https://arxiv.org/ps/1605.01693">ps</a>, <a href="https://arxiv.org/format/1605.01693">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Local Rigidity Of Uniform Lattices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Levit%2C+A">Arie Levit</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1605.01693v4-abstract-short" style="display: inline;"> We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to irreducible uniform lattices in $\text{Isom}(X)$ where $X$ is a proper $\text{CAT}(0)$ space with no Euclidian factors, not isometric to the hyperbolic plane. We deduc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.01693v4-abstract-full').style.display = 'inline'; document.getElementById('1605.01693v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.01693v4-abstract-full" style="display: none;"> We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to irreducible uniform lattices in $\text{Isom}(X)$ where $X$ is a proper $\text{CAT}(0)$ space with no Euclidian factors, not isometric to the hyperbolic plane. We deduce an analog of Wang's finiteness theorem for certain non-positively curved metric spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.01693v4-abstract-full').style.display = 'none'; document.getElementById('1605.01693v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 November, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 May, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">36 pages, no figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22D05; 22E40; 20F67 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1511.05767">arXiv:1511.05767</a> <span> [<a href="https://arxiv.org/pdf/1511.05767">pdf</a>, <a href="https://arxiv.org/format/1511.05767">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Maximal subgroups of SL(n, Z) </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Meiri%2C+C">Chen Meiri</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1511.05767v2-abstract-short" style="display: inline;"> We establish the existence of maximal subgroups of various diferent natures in SL(n,Z). In particular, we prove that there are continuously many maximal subgroups, we provide a maximal subgroup whose action on the projective space has no dense orbits, and we produce a faithful primitive permutation representation of PSL(n,Z) which is not 2-transitive. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1511.05767v2-abstract-full" style="display: none;"> We establish the existence of maximal subgroups of various diferent natures in SL(n,Z). In particular, we prove that there are continuously many maximal subgroups, we provide a maximal subgroup whose action on the projective space has no dense orbits, and we produce a faithful primitive permutation representation of PSL(n,Z) which is not 2-transitive. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.05767v2-abstract-full').style.display = 'none'; document.getElementById('1511.05767v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 November, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages, 0 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G15; 22E40; 11F06; 20H20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1510.05423">arXiv:1510.05423</a> <span> [<a href="https://arxiv.org/pdf/1510.05423">pdf</a>, <a href="https://arxiv.org/ps/1510.05423">ps</a>, <a href="https://arxiv.org/format/1510.05423">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Kazhdan-Margulis theorem for Invariant Random Subgroups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1510.05423v4-abstract-short" style="display: inline;"> Given a simple Lie group $G$, we show that the lattices in $G$ are weakly uniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem. Our proof however is straightforward --- considering general IRS rather than lattices allows us to apply a compactness argument. In terms of p.m.p. actions, we show that for every $蔚$ there is an identity neighbourhood $U$ which intersects trivially… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1510.05423v4-abstract-full').style.display = 'inline'; document.getElementById('1510.05423v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1510.05423v4-abstract-full" style="display: none;"> Given a simple Lie group $G$, we show that the lattices in $G$ are weakly uniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem. Our proof however is straightforward --- considering general IRS rather than lattices allows us to apply a compactness argument. In terms of p.m.p. actions, we show that for every $蔚$ there is an identity neighbourhood $U$ which intersects trivially the stabilizers of $1-蔚$ of the points in every non-atomic $G$-space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1510.05423v4-abstract-full').style.display = 'none'; document.getElementById('1510.05423v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 October, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">4 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1509.01711">arXiv:1509.01711</a> <span> [<a href="https://arxiv.org/pdf/1509.01711">pdf</a>, <a href="https://arxiv.org/ps/1509.01711">ps</a>, <a href="https://arxiv.org/format/1509.01711">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1215/00127094-2017-0020">10.1215/00127094-2017-0020 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Rank, combinatorial cost and homology torsion growth in higher rank lattices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Abert%2C+M">Miklos Abert</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Nikolov%2C+N">Nikolay Nikolov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1509.01711v2-abstract-short" style="display: inline;"> We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right angled groups. A group is right angled if it can be generated by a sequence… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.01711v2-abstract-full').style.display = 'inline'; document.getElementById('1509.01711v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1509.01711v2-abstract-full" style="display: none;"> We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. Most non-uniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (co-compact) right angled arithmetic groups in $\mathrm{SL}(n,\mathbb{R}),~n\geq 3$ and $\mathrm{SO}(p,q)$ for some values of $p,q$. This is a class of lattices for which the Congruence Subgroup Property is not known in general. Using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right angled lattice in a higher rank simple Lie group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.01711v2-abstract-full').style.display = 'none'; document.getElementById('1509.01711v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages, to appear in Duke Math</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20E26; 20F05; 20F69; 22E40; 57T15 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Duke Math. J. 166, no. 15 (2017), 2925-2964 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.06881">arXiv:1505.06881</a> <span> [<a href="https://arxiv.org/pdf/1505.06881">pdf</a>, <a href="https://arxiv.org/ps/1505.06881">ps</a>, <a href="https://arxiv.org/format/1505.06881">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Higher rank arithmetic groups which are not invariably generated </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Meiri%2C+C">Chen Meiri</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1505.06881v4-abstract-short" style="display: inline;"> It was conjectured in [KLS14] that for arithmetic groups, Invariable Generation is equivalent to the Congruence Subgroup Property. In view of the famous Serre conjecture this would imply that higher rank arithmetic groups are invariably generated. In this paper we prove that some higher rank arithmetic groups are not invariably generated. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.06881v4-abstract-full" style="display: none;"> It was conjectured in [KLS14] that for arithmetic groups, Invariable Generation is equivalent to the Congruence Subgroup Property. In view of the famous Serre conjecture this would imply that higher rank arithmetic groups are invariably generated. In this paper we prove that some higher rank arithmetic groups are not invariably generated. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.06881v4-abstract-full').style.display = 'none'; document.getElementById('1505.06881v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G15; 22E40; 11F06; 20H20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.00415">arXiv:1505.00415</a> <span> [<a href="https://arxiv.org/pdf/1505.00415">pdf</a>, <a href="https://arxiv.org/ps/1505.00415">ps</a>, <a href="https://arxiv.org/format/1505.00415">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Infinitesimal generators and quasi non-archimedean topological groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Ma%C3%AEtre%2C+F+L">Fran莽ois Le Ma卯tre</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1505.00415v2-abstract-short" style="display: inline;"> We show that connected separable locally compact groups are infinitesimally finitely generated, meaning that there is an integer $n$ such that every neighborhood of the identity contains $n$ elements generating a dense subgroup. We generalize a theorem of Schreier and Ulam by showing that any separable connected compact group is infinitesimally $2$-generated. Inspired by a result of Kechris, we… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.00415v2-abstract-full').style.display = 'inline'; document.getElementById('1505.00415v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.00415v2-abstract-full" style="display: none;"> We show that connected separable locally compact groups are infinitesimally finitely generated, meaning that there is an integer $n$ such that every neighborhood of the identity contains $n$ elements generating a dense subgroup. We generalize a theorem of Schreier and Ulam by showing that any separable connected compact group is infinitesimally $2$-generated. Inspired by a result of Kechris, we introduce the notion of a quasi non-archimedean group. We observe that full groups are quasi non-archimedean, and that every continuous homomorphism from an infinitesimally finitely generated group into a quasi non-archimedean group is trivial. We prove that a locally compact group is quasi non-archimedean if and only if it is totally disconnected, and provide various examples which show that the picture is much richer for Polish groups. In particular, we get an example of a Polish group which is infinitesimally $1$-generated but totally disconnected, strengthening Stevens' negative answer to Problem 160 from the Scottish book. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.00415v2-abstract-full').style.display = 'none'; document.getElementById('1505.00415v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Changed terminology and reworked the introduction; added the existence of an infinitesimally 1-generated totally disconnected Polish group. Comments welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1504.05873">arXiv:1504.05873</a> <span> [<a href="https://arxiv.org/pdf/1504.05873">pdf</a>, <a href="https://arxiv.org/ps/1504.05873">ps</a>, <a href="https://arxiv.org/format/1504.05873">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> </div> </div> <p class="title is-5 mathjax"> An Upper bound on the growth of Dirichlet tilings of hyperbolic spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Benjamini%2C+I">Itai Benjamini</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1504.05873v2-abstract-short" style="display: inline;"> It is shown that the growth rate $(\lim_r |B(r)|^{1/r})$ of any $k$ faces Dirichlet tiling of the real hyperbolic space $\mathbb{H}^d, d>2,$ is at most $k-1-蔚$, for an $蔚> 0$, depending only on $k$ and $d$. We don't know if there is a universal $蔚_u > 0$, such that $k-1-蔚_u$ upperbounds the growth rate for any $k$-regular tiling, when $ d > 2$? </span> <span class="abstract-full has-text-grey-dark mathjax" id="1504.05873v2-abstract-full" style="display: none;"> It is shown that the growth rate $(\lim_r |B(r)|^{1/r})$ of any $k$ faces Dirichlet tiling of the real hyperbolic space $\mathbb{H}^d, d>2,$ is at most $k-1-蔚$, for an $蔚> 0$, depending only on $k$ and $d$. We don't know if there is a universal $蔚_u > 0$, such that $k-1-蔚_u$ upperbounds the growth rate for any $k$-regular tiling, when $ d > 2$? <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.05873v2-abstract-full').style.display = 'none'; document.getElementById('1504.05873v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 April, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1503.08402">arXiv:1503.08402</a> <span> [<a href="https://arxiv.org/pdf/1503.08402">pdf</a>, <a href="https://arxiv.org/format/1503.08402">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> A lecture on Invariant Random Subgroups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1503.08402v2-abstract-short" style="display: inline;"> Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups in a given group G. They can be regarded both as a generalization of normal subgroups as well as a generalization of lattices. As such, it is intriguing to extend results from the theories of normal subgroups and of lattices to the context of IRS. Another approach is to analyse and then use the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.08402v2-abstract-full').style.display = 'inline'; document.getElementById('1503.08402v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1503.08402v2-abstract-full" style="display: none;"> Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups in a given group G. They can be regarded both as a generalization of normal subgroups as well as a generalization of lattices. As such, it is intriguing to extend results from the theories of normal subgroups and of lattices to the context of IRS. Another approach is to analyse and then use the space IRS(G) as a compact G-space in order to establish new results about lattices. The second approach has been taken in the work [7s12], that came to be known as the seven samurai paper. In these lecture notes we shall try to give a taste of both approaches. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.08402v2-abstract-full').style.display = 'none'; document.getElementById('1503.08402v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 October, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, 5 figures. Based on a mini course given in Ghys' birthday --- conference Geometries in Action (2015), Oberwolfach workshop on Locally Compact Groups (2014) and Ventotene's conference on Manifolds and Groups (2015)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22Dxx; 22Exx </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1408.4217">arXiv:1408.4217</a> <span> [<a href="https://arxiv.org/pdf/1408.4217">pdf</a>, <a href="https://arxiv.org/ps/1408.4217">ps</a>, <a href="https://arxiv.org/format/1408.4217">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> Equicontinuous actions of semisimple groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bader%2C+U">Uri Bader</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1408.4217v4-abstract-short" style="display: inline;"> We study equicontinuous actions of semisimple groups and some generalizations. We prove that any such action is universally closed, and in particular proper. We derive various applications, both old and new, including closedness of continuous homomorphisms, nonexistence of weaker topologies, metric ergodicity of transitive actions and vanishing of matrix coefficients for reflexive (more generally:… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.4217v4-abstract-full').style.display = 'inline'; document.getElementById('1408.4217v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1408.4217v4-abstract-full" style="display: none;"> We study equicontinuous actions of semisimple groups and some generalizations. We prove that any such action is universally closed, and in particular proper. We derive various applications, both old and new, including closedness of continuous homomorphisms, nonexistence of weaker topologies, metric ergodicity of transitive actions and vanishing of matrix coefficients for reflexive (more generally: WAP) representations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.4217v4-abstract-full').style.display = 'none'; document.getElementById('1408.4217v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 August, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages. The introduction has been improved. (We also extended the discussion about week topologies.)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E46; 54E15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1407.7226">arXiv:1407.7226</a> <span> [<a href="https://arxiv.org/pdf/1407.7226">pdf</a>, <a href="https://arxiv.org/ps/1407.7226">ps</a>, <a href="https://arxiv.org/format/1407.7226">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Convergence groups are not invariably generated </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1407.7226v3-abstract-short" style="display: inline;"> It was conjectured in [KLS14] that non-elementary word hyperbolic groups are never invariably generated. We show that this is indeed the case even for the much larger class of convergence groups. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1407.7226v3-abstract-full" style="display: none;"> It was conjectured in [KLS14] that non-elementary word hyperbolic groups are never invariably generated. We show that this is indeed the case even for the much larger class of convergence groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1407.7226v3-abstract-full').style.display = 'none'; document.getElementById('1407.7226v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 July, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">7 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20F67; 22E40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1407.2872">arXiv:1407.2872</a> <span> [<a href="https://arxiv.org/pdf/1407.2872">pdf</a>, <a href="https://arxiv.org/format/1407.2872">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> Invariant random subgroups of linear groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Glasner%2C+Y">Yair Glasner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1407.2872v3-abstract-short" style="display: inline;"> Let $螕< \mathrm{GL}_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure, $\mathrm{Sub}(螕)$ the space of all subgroups of $螕$ with the, compact, metric, Chabauty topology. An invariant random subgroup (IRS) of $螕$ is a conjugation invariant Borel probability measure on $\mathrm{Sub}(螕)$. An $\mathrm{IRS}$ is called nontrivial if it does not have an atom in the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1407.2872v3-abstract-full').style.display = 'inline'; document.getElementById('1407.2872v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1407.2872v3-abstract-full" style="display: none;"> Let $螕< \mathrm{GL}_n(F)$ be a countable non-amenable linear group with a simple, center free Zariski closure, $\mathrm{Sub}(螕)$ the space of all subgroups of $螕$ with the, compact, metric, Chabauty topology. An invariant random subgroup (IRS) of $螕$ is a conjugation invariant Borel probability measure on $\mathrm{Sub}(螕)$. An $\mathrm{IRS}$ is called nontrivial if it does not have an atom in the trivial group, i.e. if it is nontrivial almost surely. We denote by $\mathrm{IRS}^{0}(螕)$ the collection of all nontrivial $\mathrm{IRS}$ on $螕$. We show that there exits a free subgroup $F < 螕$ and a non-discrete group topology $\mathrm{St}$ on $螕$ such that for every $渭\in \mathrm{IRS}^{0}(螕)$ the following properties hold: (i) $渭$-almost every subgroup of $螕$ is open. (ii) $F \cdot 螖= 螕$ for $渭$-almost every $螖\in \mathrm{Sub}(螕)$. (iii) $F \cap 螖$ is infinitely generated, for every open subgroup. (iv) The map $桅: (\mathrm{Sub}(螕),渭) \rightarrow (\mathrm{Sub}(F),桅_* 渭)$ given by $螖\mapsto 螖\cap F$, is an $F$-invariant isomorphism of probability spaces. We say that an action of $螕$ on a probability space, by measure preserving transformations, is almost surely non free (ASNF) if almost all point stabilizers are non-trivial. As a corollary of the above theorem we show that the product of finitely many ANSF $螕$-spaces, with the diagonal $螕$ action, is ASNF. Let $螕< \mathrm{GL}_n(F)$ be a countable linear group, $A \lhd 螕$ the maximal normal amenable subgroup of $螕$. We show that if $渭\in \mathrm{IRS}(螕)$ is supported on amenable subgroups of $螕$ then in fact it is supported on $\mathrm{Sub}(A)$. In particular if $A(螕) = \langle e \rangle$ then $螖= \langle e \rangle, 渭$ almost surely. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1407.2872v3-abstract-full').style.display = 'none'; document.getElementById('1407.2872v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 July, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Main article by Yair Glasner with an appendix by Tsachik Gelander and Yair Glasner. 41 pages 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 28D15; Secondary 34A20; 20E05; 20E25; 20E42 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1402.0962">arXiv:1402.0962</a> <span> [<a href="https://arxiv.org/pdf/1402.0962">pdf</a>, <a href="https://arxiv.org/ps/1402.0962">ps</a>, <a href="https://arxiv.org/format/1402.0962">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Lectures on Lattices and locally symmetric spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1402.0962v1-abstract-short" style="display: inline;"> The aim of this short lecture series is to expose the students to the beautiful theory of lattices by, on one hand, demonstrating various basic ideas that appear in this theory and, on the other hand, formulating some of the celebrated results which, in particular, shows some connections to other fields of mathematics. The time restriction forces us to avoid many important parts of the theory, and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1402.0962v1-abstract-full').style.display = 'inline'; document.getElementById('1402.0962v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1402.0962v1-abstract-full" style="display: none;"> The aim of this short lecture series is to expose the students to the beautiful theory of lattices by, on one hand, demonstrating various basic ideas that appear in this theory and, on the other hand, formulating some of the celebrated results which, in particular, shows some connections to other fields of mathematics. The time restriction forces us to avoid many important parts of the theory, and the route we have chosen is naturally biased by the individual taste of the speaker. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1402.0962v1-abstract-full').style.display = 'none'; document.getElementById('1402.0962v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 February, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Lecture notes of a mini course given in PCMI at July 2012</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40; 57N16; 20G20; 20E07; 53C30; 53C35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1401.8003">arXiv:1401.8003</a> <span> [<a href="https://arxiv.org/pdf/1401.8003">pdf</a>, <a href="https://arxiv.org/ps/1401.8003">ps</a>, <a href="https://arxiv.org/format/1401.8003">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Counting commensurability classes of hyperbolic manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Levit%2C+A">Arie Levit</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1401.8003v3-abstract-short" style="display: inline;"> Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Our met… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1401.8003v3-abstract-full').style.display = 'inline'; document.getElementById('1401.8003v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1401.8003v3-abstract-full" style="display: none;"> Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1401.8003v3-abstract-full').style.display = 'none'; document.getElementById('1401.8003v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 May, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 January, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">version to appear in Geometric and Functional Analysis</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40; 57N16; 20G20; 20E07; 53C30; 53C35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1303.1161">arXiv:1303.1161</a> <span> [<a href="https://arxiv.org/pdf/1303.1161">pdf</a>, <a href="https://arxiv.org/ps/1303.1161">ps</a>, <a href="https://arxiv.org/format/1303.1161">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/blms/bdt063">10.1112/blms/bdt063 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Word Values in p-Adic and Adelic Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Avni%2C+N">Nir Avni</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Kassabov%2C+M">Martin Kassabov</a>, <a href="/search/math?searchtype=author&query=Shalev%2C+A">Aner Shalev</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1303.1161v1-abstract-short" style="display: inline;"> We study the sets of values of words in p-adic and adelic groups. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1303.1161v1-abstract-full" style="display: none;"> We study the sets of values of words in p-adic and adelic groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1303.1161v1-abstract-full').style.display = 'none'; document.getElementById('1303.1161v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 March, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G25 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1210.2961">arXiv:1210.2961</a> <span> [<a href="https://arxiv.org/pdf/1210.2961">pdf</a>, <a href="https://arxiv.org/ps/1210.2961">ps</a>, <a href="https://arxiv.org/format/1210.2961">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="K-Theory and Homology">math.KT</span> </div> </div> <p class="title is-5 mathjax"> On the growth of $L^2$-invariants for sequences of lattices in Lie groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Abert%2C+M">Miklos Abert</a>, <a href="/search/math?searchtype=author&query=Bergeron%2C+N">Nicolas Bergeron</a>, <a href="/search/math?searchtype=author&query=Biringer%2C+I">Ian Biringer</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Nikolov%2C+N">Nikolay Nikolov</a>, <a href="/search/math?searchtype=author&query=Raimbault%2C+J">Jean Raimbault</a>, <a href="/search/math?searchtype=author&query=Samet%2C+I">Iddo Samet</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1210.2961v4-abstract-short" style="display: inline;"> We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion of Benjamini--Schramm convergence (BS-convergence), originally introduced for s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.2961v4-abstract-full').style.display = 'inline'; document.getElementById('1210.2961v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1210.2961v4-abstract-full" style="display: none;"> We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion of Benjamini--Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces implies convergence, in an appropriate sense, of the associated normalized relative Plancherel measures. This then yields convergence of normalized multiplicities of unitary representations, Betti numbers and other spectral invariants. On the other hand, when the corresponding Lie group $G$ is simple and of real rank at least two, we prove that there is only one possible BS-limit, i.e. when the volume tends to infinity, locally symmetric spaces always BS-converge to their universal cover $G/K$. This leads to various general uniform results. When restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of BS-convergence which in turn implies upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak--Xue. An important role in our approach is played by the notion of Invariant Random Subgroups. For higher rank simple Lie groups $G$, we exploit rigidity theory, and in particular the Nevo--St眉ck--Zimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRSs of $G$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1210.2961v4-abstract-full').style.display = 'none'; document.getElementById('1210.2961v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 January, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 October, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">The first version of this paper has been split into two papers. This is the first part. It is 64 pages long. To appear in Annals of Mathematics</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1205.6553">arXiv:1205.6553</a> <span> [<a href="https://arxiv.org/pdf/1205.6553">pdf</a>, <a href="https://arxiv.org/ps/1205.6553">ps</a>, <a href="https://arxiv.org/format/1205.6553">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> Limits of finite homogeneous metric spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1205.6553v2-abstract-short" style="display: inline;"> We classify the metric spaces that can be approximated by finite homogeneous ones. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1205.6553v2-abstract-full" style="display: none;"> We classify the metric spaces that can be approximated by finite homogeneous ones. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1205.6553v2-abstract-full').style.display = 'none'; document.getElementById('1205.6553v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 March, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 May, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20E18; 22C05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1205.1140">arXiv:1205.1140</a> <span> [<a href="https://arxiv.org/pdf/1205.1140">pdf</a>, <a href="https://arxiv.org/ps/1205.1140">ps</a>, <a href="https://arxiv.org/format/1205.1140">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Homogeneous number of free generators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Aka%2C+M">Menny Aka</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Soifer%2C+G+A">Gregory A. Soifer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1205.1140v1-abstract-short" style="display: inline;"> We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group 螕which admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1205.1140v1-abstract-full').style.display = 'inline'; document.getElementById('1205.1140v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1205.1140v1-abstract-full" style="display: none;"> We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group 螕which admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group 螕is uncountable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1205.1140v1-abstract-full').style.display = 'none'; document.getElementById('1205.1140v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 May, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2012. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1201.0332">arXiv:1201.0332</a> <span> [<a href="https://arxiv.org/pdf/1201.0332">pdf</a>, <a href="https://arxiv.org/ps/1201.0332">ps</a>, <a href="https://arxiv.org/format/1201.0332">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Topology">math.AT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/agt.2013.13.1733">10.2140/agt.2013.13.1733 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Milnor-Wood inequalities for products </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bucher%2C+M">Michelle Bucher</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1201.0332v1-abstract-short" style="display: inline;"> We prove Milnor-Wood inequalities for local products of manifolds. As a consequence, we establish the generalized Chern Conjecture for products $M\times 危^k$ for any product of a manifold $M$ with a product of $k$ copies of a surface $危$ for $k$ sufficiently large. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1201.0332v1-abstract-full" style="display: none;"> We prove Milnor-Wood inequalities for local products of manifolds. As a consequence, we establish the generalized Chern Conjecture for products $M\times 危^k$ for any product of a manifold $M$ with a product of $k$ copies of a surface $危$ for $k$ sufficiently large. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1201.0332v1-abstract-full').style.display = 'none'; document.getElementById('1201.0332v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 January, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages; MSC keywords: characteristic classes and numbers</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Algebr. Geom. Topol. 13 (2013) 1733-1742 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1104.5559">arXiv:1104.5559</a> <span> [<a href="https://arxiv.org/pdf/1104.5559">pdf</a>, <a href="https://arxiv.org/ps/1104.5559">ps</a>, <a href="https://arxiv.org/format/1104.5559">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> On the growth of Betti numbers of locally symmetric spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Abert%2C+M">Miklos Abert</a>, <a href="/search/math?searchtype=author&query=Bergeron%2C+N">Nicolas Bergeron</a>, <a href="/search/math?searchtype=author&query=Biringer%2C+I">Ian Biringer</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Nikolov%2C+N">Nikolay Nikolov</a>, <a href="/search/math?searchtype=author&query=Raimbault%2C+J">Jean Raimbault</a>, <a href="/search/math?searchtype=author&query=Samet%2C+I">Iddo Samet</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1104.5559v1-abstract-short" style="display: inline;"> We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the L眉ck Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gromov in \cite{BGS}. The basic idea is to adapt the theory of local convergence… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1104.5559v1-abstract-full').style.display = 'inline'; document.getElementById('1104.5559v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1104.5559v1-abstract-full" style="display: none;"> We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the L眉ck Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gromov in \cite{BGS}. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1104.5559v1-abstract-full').style.display = 'none'; document.getElementById('1104.5559v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 April, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Announcement</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1104.4774">arXiv:1104.4774</a> <span> [<a href="https://arxiv.org/pdf/1104.4774">pdf</a>, <a href="https://arxiv.org/ps/1104.4774">ps</a>, <a href="https://arxiv.org/format/1104.4774">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> The dynamics of Aut(F_n) on redundant representations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Minsky%2C+Y">Yair Minsky</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1104.4774v2-abstract-short" style="display: inline;"> We study some dynamical properties of the canonical Aut(F_n)-action on the space R_n(G) of redundant representations of the free group F_n in G, where G is the group of rational points of a simple algebraic group over a local field. We show that this action is always minimal and ergodic, confirming a conjecture of A. Lubotzky. On the other hand for the classical cases where G=SL(2,R) or SL(2,C) we… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1104.4774v2-abstract-full').style.display = 'inline'; document.getElementById('1104.4774v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1104.4774v2-abstract-full" style="display: none;"> We study some dynamical properties of the canonical Aut(F_n)-action on the space R_n(G) of redundant representations of the free group F_n in G, where G is the group of rational points of a simple algebraic group over a local field. We show that this action is always minimal and ergodic, confirming a conjecture of A. Lubotzky. On the other hand for the classical cases where G=SL(2,R) or SL(2,C) we show that the action is not weak mixing, in the sense that the diagonal action on R_n(G)^2 is not ergodic. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1104.4774v2-abstract-full').style.display = 'none'; document.getElementById('1104.4774v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 April, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Some of the statements and arguments rely on the assumption that the algebraic group G is simply connected. This assumption, which was missing in the previous version, is not necessary in the archimedean cases, but it is needed in the non-archimedean cases</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22D40; 57M50 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Groups Geom. Dynamics, 7 (2013), no. 3, 557-576 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1102.3574">arXiv:1102.3574</a> <span> [<a href="https://arxiv.org/pdf/1102.3574">pdf</a>, <a href="https://arxiv.org/ps/1102.3574">ps</a>, <a href="https://arxiv.org/format/1102.3574">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Volume vs. rank of lattices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1102.3574v1-abstract-short" style="display: inline;"> We show the rank (i.e. minimal size of a generating set) of lattices cannot grow faster than the volume. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1102.3574v1-abstract-full" style="display: none;"> We show the rank (i.e. minimal size of a generating set) of lattices cannot grow faster than the volume. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1102.3574v1-abstract-full').style.display = 'none'; document.getElementById('1102.3574v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 February, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40 Discrete subgroups of Lie groups </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1012.1488">arXiv:1012.1488</a> <span> [<a href="https://arxiv.org/pdf/1012.1488">pdf</a>, <a href="https://arxiv.org/ps/1012.1488">ps</a>, <a href="https://arxiv.org/format/1012.1488">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00222-011-0363-2">10.1007/s00222-011-0363-2 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A fixed point theorem for L^1 spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bader%2C+U">Uri Bader</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Monod%2C+N">Nicolas Monod</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1012.1488v2-abstract-short" style="display: inline;"> We prove a fixed point theorem for a family of Banach spaces, notably L^1 and its non-commutative analogues. Several applications are given, e.g. the optimal solution to the "derivation problem" studied since the 1960s. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1012.1488v2-abstract-full" style="display: none;"> We prove a fixed point theorem for a family of Banach spaces, notably L^1 and its non-commutative analogues. Several applications are given, e.g. the optimal solution to the "derivation problem" studied since the 1960s. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1012.1488v2-abstract-full').style.display = 'none'; document.getElementById('1012.1488v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 October, 2011; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 December, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">minor additions; to appear in Inv. Math</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Inventiones Math. 189 No. 1 (2012), 143--148 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1008.2911">arXiv:1008.2911</a> <span> [<a href="https://arxiv.org/pdf/1008.2911">pdf</a>, <a href="https://arxiv.org/ps/1008.2911">ps</a>, <a href="https://arxiv.org/format/1008.2911">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/blms/bdr061">10.1112/blms/bdr061 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Simple groups without lattices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bader%2C+U">Uri Bader</a>, <a href="/search/math?searchtype=author&query=Caprace%2C+P">Pierre-Emmanuel Caprace</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Mozes%2C+S">Shahar Mozes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1008.2911v2-abstract-short" style="display: inline;"> We show that the group of almost automorphisms of a d-regular tree does not admit lattices. As far as we know this is the first such example among (compactly generated) simple locally compact groups. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1008.2911v2-abstract-full" style="display: none;"> We show that the group of almost automorphisms of a d-regular tree does not admit lattices. As far as we know this is the first such example among (compactly generated) simple locally compact groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1008.2911v2-abstract-full').style.display = 'none'; document.getElementById('1008.2911v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 March, 2011; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 August, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages. Revised according to referee's report</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40; 20E08 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Bull. Lond. Math. Soc. 44 (2012), no. 1, 55-67 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0911.0628">arXiv:0911.0628</a> <span> [<a href="https://arxiv.org/pdf/0911.0628">pdf</a>, <a href="https://arxiv.org/ps/0911.0628">ps</a>, <a href="https://arxiv.org/format/0911.0628">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> An Aschbacher--O'Nan--Scott theorem for countable linear groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Glasner%2C+Y">Yair Glasner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0911.0628v3-abstract-short" style="display: inline;"> The purpose of this note is to extend the classical Aschbacher--O'Nan--Scott theorem for finite groups to the class of countable linear groups. This relies on the analysis of primitive actions carried out in a previous paper. Unlike the situation for finite groups, we show here that the number of primitive actions depends on the type: linear groups of almost simple type admit infinitely (and in fa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0911.0628v3-abstract-full').style.display = 'inline'; document.getElementById('0911.0628v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0911.0628v3-abstract-full" style="display: none;"> The purpose of this note is to extend the classical Aschbacher--O'Nan--Scott theorem for finite groups to the class of countable linear groups. This relies on the analysis of primitive actions carried out in a previous paper. Unlike the situation for finite groups, we show here that the number of primitive actions depends on the type: linear groups of almost simple type admit infinitely (and in fact unaccountably) many primitive actions, while affine and diagonal groups admit only one. The abundance of primitive permutation representations is particularly interesting for rigid groups such as simple and arithmetic ones. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0911.0628v3-abstract-full').style.display = 'none'; document.getElementById('0911.0628v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 March, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 November, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 20B15; Secondary 20B10; 20B07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0904.3771">arXiv:0904.3771</a> <span> [<a href="https://arxiv.org/pdf/0904.3771">pdf</a>, <a href="https://arxiv.org/format/0904.3771">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11854-010-0030-3">10.1007/s11854-010-0030-3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Compactifications and algebraic completions of Limit groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Barlev%2C+J">Jonathan Barlev</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0904.3771v3-abstract-short" style="display: inline;"> In this paper we consider the existence of dense embeddings of Limit groups in locally compact groups generalizing earlier work of Breuillard, Gelander, Souto and Storm [GBSS] where surface groups were considered. Our main results are proved in the context of compact groups and algebraic groups over local fields. In addition we prove a generalization of the classical Baumslag lemma which is a usef… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0904.3771v3-abstract-full').style.display = 'inline'; document.getElementById('0904.3771v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0904.3771v3-abstract-full" style="display: none;"> In this paper we consider the existence of dense embeddings of Limit groups in locally compact groups generalizing earlier work of Breuillard, Gelander, Souto and Storm [GBSS] where surface groups were considered. Our main results are proved in the context of compact groups and algebraic groups over local fields. In addition we prove a generalization of the classical Baumslag lemma which is a useful tool for generating eventually faithful sequences of homomorphisms. The last section is dedicated to correct a mistake from [BGSS] and to get rid of the even genus assumption. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0904.3771v3-abstract-full').style.display = 'none'; document.getElementById('0904.3771v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 February, 2011; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 April, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">v2: Substantial changes to sections 7 and 8.2. Typos corrected. References added. v3: Acknowledgement corrected</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal D'Analyse Mathematique Volume 112 (2010), Number 1, 261-287 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0902.1215">arXiv:0902.1215</a> <span> [<a href="https://arxiv.org/pdf/0902.1215">pdf</a>, <a href="https://arxiv.org/ps/0902.1215">ps</a>, <a href="https://arxiv.org/format/0902.1215">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> The generalized Chern conjecture for manifolds that are locally a product of surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bucher%2C+M">Michelle Bucher</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0902.1215v3-abstract-short" style="display: inline;"> We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old conjecture proved by Milnor in dimension 2. In particular, the Chern conjecture follows in these cases. The proof goes via a new sharp Milnor--Wood inequality for Ri… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0902.1215v3-abstract-full').style.display = 'inline'; document.getElementById('0902.1215v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0902.1215v3-abstract-full" style="display: none;"> We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old conjecture proved by Milnor in dimension 2. In particular, the Chern conjecture follows in these cases. The proof goes via a new sharp Milnor--Wood inequality for Riemannian manifolds that are locally a product of hyperbolic planes. Furthermore, we analyze the possible flat vector bundles over such manifolds. Over closed Hilbert--Blumenthal modular varieties, we show that there are finitely many flat structures with nonzero Euler number and none of them corresponds to the tangent bundle. Some of the main results were announced in C.R. Acad. Sci. Paris, Ser. I 346 (2008) 661-666. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0902.1215v3-abstract-full').style.display = 'none'; document.getElementById('0902.1215v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 May, 2009; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 February, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">36 pages. New title, modified abstract and introduction</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R20; 53C35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0811.2482">arXiv:0811.2482</a> <span> [<a href="https://arxiv.org/pdf/0811.2482">pdf</a>, <a href="https://arxiv.org/ps/0811.2482">ps</a>, <a href="https://arxiv.org/format/0811.2482">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Counting arithmetic lattices and surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Belolipetsky%2C+M">Mikhail Belolipetsky</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a>, <a href="/search/math?searchtype=author&query=Lubotzky%2C+A">Alex Lubotzky</a>, <a href="/search/math?searchtype=author&query=Shalev%2C+A">Aner Shalev</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0811.2482v2-abstract-short" style="display: inline;"> We give estimates on the number $AL_H(x)$ of arithmetic lattices $螕$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use several different te… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0811.2482v2-abstract-full').style.display = 'inline'; document.getElementById('0811.2482v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0811.2482v2-abstract-full" style="display: none;"> We give estimates on the number $AL_H(x)$ of arithmetic lattices $螕$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $螕$ as a function of its covolume), number theoretic (bounding the number of maximal such $螕$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $螕$). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0811.2482v2-abstract-full').style.display = 'none'; document.getElementById('0811.2482v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 April, 2010; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 November, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages, final version, to appear in Annals of Mathematics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E40 (57N16; 20G20; 20H10; 20C30; 20E07) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0805.3813">arXiv:0805.3813</a> <span> [<a href="https://arxiv.org/pdf/0805.3813">pdf</a>, <a href="https://arxiv.org/ps/0805.3813">ps</a>, <a href="https://arxiv.org/format/0805.3813">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> </div> </div> <p class="title is-5 mathjax"> On fixed points and uniformly convex spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=gelander%2C+t">tsachik gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0805.3813v1-abstract-short" style="display: inline;"> The purpose of this note is to present two elementary, but useful, facts concerning actions on uniformly convex spaces. We demonstrate how each of them can be used in an alternative proof of the triviality of the first $L_p$-cohomology of higher rank simple Lie groups, proved in [BFGM]. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0805.3813v1-abstract-full" style="display: none;"> The purpose of this note is to present two elementary, but useful, facts concerning actions on uniformly convex spaces. We demonstrate how each of them can be used in an alternative proof of the triviality of the first $L_p$-cohomology of higher rank simple Lie groups, proved in [BFGM]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0805.3813v1-abstract-full').style.display = 'none'; document.getElementById('0805.3813v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 May, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">2 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0804.1997">arXiv:0804.1997</a> <span> [<a href="https://arxiv.org/pdf/0804.1997">pdf</a>, <a href="https://arxiv.org/ps/0804.1997">ps</a>, <a href="https://arxiv.org/format/0804.1997">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Milnor-Wood inequalities for manifolds locally isometric to a product of hyperbolic planes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bucher%2C+M">Michelle Bucher</a>, <a href="/search/math?searchtype=author&query=Gelander%2C+T">Tsachik Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0804.1997v1-abstract-short" style="display: inline;"> This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0804.1997v1-abstract-full').style.display = 'inline'; document.getElementById('0804.1997v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0804.1997v1-abstract-full" style="display: none;"> This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to many other locally symmetric spaces they do admit flat vector bundle of the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0804.1997v1-abstract-full').style.display = 'none'; document.getElementById('0804.1997v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 April, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">6 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R20; 53C35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0803.3383">arXiv:0803.3383</a> <span> [<a href="https://arxiv.org/pdf/0803.3383">pdf</a>, <a href="https://arxiv.org/ps/0803.3383">ps</a>, <a href="https://arxiv.org/format/0803.3383">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Notes on uniform exponential growth and Tits alternative </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">T. Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0803.3383v1-abstract-short" style="display: inline;"> These notes contain results concerning uniform exponential growth which were obtained in collaborations with E. Breuillard and A. Salehi-Golsefidy, mostly during 2005, improving Eskin-Mozes-Oh theorem \cite{EMO}, as well as a uniform uniform version of Tits alternative improving \cite{uti}. </span> <span class="abstract-full has-text-grey-dark mathjax" id="0803.3383v1-abstract-full" style="display: none;"> These notes contain results concerning uniform exponential growth which were obtained in collaborations with E. Breuillard and A. Salehi-Golsefidy, mostly during 2005, improving Eskin-Mozes-Oh theorem \cite{EMO}, as well as a uniform uniform version of Tits alternative improving \cite{uti}. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0803.3383v1-abstract-full').style.display = 'none'; document.getElementById('0803.3383v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 March, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2008. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0611830">arXiv:math/0611830</a> <span> [<a href="https://arxiv.org/pdf/math/0611830">pdf</a>, <a href="https://arxiv.org/ps/math/0611830">ps</a>, <a href="https://arxiv.org/format/math/0611830">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> On deformations of free groups in compact Lie groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gelander%2C+T">T. Gelander</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="math/0611830v2-abstract-short" style="display: inline;"> We study some properties of the varieties of deformations of free groups in compact Lie groups. In particular we prove a conjecture of Margulis and Soifer about the density of non-virtually free points in such variety, and a conjecture of Goldman on the ergodicity of the action of Aut(Fn) on such variety when n>2. </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0611830v2-abstract-full" style="display: none;"> We study some properties of the varieties of deformations of free groups in compact Lie groups. In particular we prove a conjecture of Margulis and Soifer about the density of non-virtually free points in such variety, and a conjecture of Goldman on the ergodicity of the action of Aut(Fn) on such variety when n>2. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0611830v2-abstract-full').style.display = 'none'; document.getElementById('math/0611830v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 November, 2006; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2006. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F28 </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" 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