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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Bon, N M"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <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/2410.02614">arXiv:2410.02614</a> <span> [<a href="https://arxiv.org/pdf/2410.02614">pdf</a>, <a href="https://arxiv.org/ps/2410.02614">ps</a>, <a href="https://arxiv.org/format/2410.02614">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"> Subexponential growth and $C^1$ actions on one-manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Kim%2C+S">Sang-hyun Kim</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=de+la+Salle%2C+M">Mikael de la Salle</a>, <a href="/search/math?searchtype=author&query=Triestino%2C+M">Michele Triestino</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="2410.02614v1-abstract-short" style="display: inline;"> Let $G$ be a countable group with no finitely generated subgroup of exponential growth. We show that every action of $G$ on a countable set preserving a linear (respectively, circular) order can be realised as the restriction of some action by $C^1$ diffeomorphisms on an interval (respectively, the circle) to an invariant subset. As a consequence, every action of $G$ by homeomorphisms on a compact… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.02614v1-abstract-full').style.display = 'inline'; document.getElementById('2410.02614v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.02614v1-abstract-full" style="display: none;"> Let $G$ be a countable group with no finitely generated subgroup of exponential growth. We show that every action of $G$ on a countable set preserving a linear (respectively, circular) order can be realised as the restriction of some action by $C^1$ diffeomorphisms on an interval (respectively, the circle) to an invariant subset. As a consequence, every action of $G$ by homeomorphisms on a compact connected one-manifold can be made $C^1$ upon passing to a semi-conjugate action. The proof is based on a functional characterisation of groups of local subexponential growth. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.02614v1-abstract-full').style.display = 'none'; document.getElementById('2410.02614v1-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> 3 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">14 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/2401.06886">arXiv:2401.06886</a> <span> [<a href="https://arxiv.org/pdf/2401.06886">pdf</a>, <a href="https://arxiv.org/ps/2401.06886">ps</a>, <a href="https://arxiv.org/format/2401.06886">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 the growth of actions of free products </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boudec%2C+A+L">Adrien Le Boudec</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Salo%2C+V">Ville Salo</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="2401.06886v1-abstract-short" style="display: inline;"> If $G$ is a finitely generated group and $X$ a $G$-set, the growth of the action of $G$ on $X$ is the function that measures the largest cardinality of a ball of radius $n$ in the Schreier graph $螕(G,X)$. In this note we consider the following stability problem: if $G,H$ are finitely generated groups admitting a faithful action of growth bounded above by a function $f$, does the free product… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.06886v1-abstract-full').style.display = 'inline'; document.getElementById('2401.06886v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.06886v1-abstract-full" style="display: none;"> If $G$ is a finitely generated group and $X$ a $G$-set, the growth of the action of $G$ on $X$ is the function that measures the largest cardinality of a ball of radius $n$ in the Schreier graph $螕(G,X)$. In this note we consider the following stability problem: if $G,H$ are finitely generated groups admitting a faithful action of growth bounded above by a function $f$, does the free product $G \ast H$ also admit a faithful action of growth bounded above by $f$? We show that the answer is positive under additional assumptions, and negative in general. In the negative direction, our counter-examples are obtained with $G$ either the commutator subgroup of the topological full group of a minimal and expansive homeomorphism of the Cantor space; or $G$ a Houghton group. In both cases, the group $G$ admits a faithful action of linear growth, and we show that $G\ast H$ admits no faithful action of subquadratic growth provided $H$ is non-trivial. In the positive direction, we describe a class of groups that admit actions of linear growth and is closed under free products and exhibit examples within this class, among which the Grigorchuk group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.06886v1-abstract-full').style.display = 'none'; document.getElementById('2401.06886v1-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 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.04207">arXiv:2310.04207</a> <span> [<a href="https://arxiv.org/pdf/2310.04207">pdf</a>, <a href="https://arxiv.org/format/2310.04207">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"> GAGTA 2023 Problem Session </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bradford%2C+H">Henry Bradford</a>, <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Fournier-Facio%2C+F">Francesco Fournier-Facio</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Petyt%2C+H">Harry Petyt</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="2310.04207v1-abstract-short" style="display: inline;"> The conference `Geometric and Asymptotic Group Theory with Applications (GAGTA) 2023: Groups and Dynamics' took place at the Erwin Schr枚dinger Institute on July 17-21. These are the problems that were proposed during the Problem Session: Residual finiteness growth, Geometric v. random walk boundaries, Stability, Cogrowth, Embedding left orderable groups, Schreier growth gap, $\ell^p$ models for hy… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.04207v1-abstract-full').style.display = 'inline'; document.getElementById('2310.04207v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.04207v1-abstract-full" style="display: none;"> The conference `Geometric and Asymptotic Group Theory with Applications (GAGTA) 2023: Groups and Dynamics' took place at the Erwin Schr枚dinger Institute on July 17-21. These are the problems that were proposed during the Problem Session: Residual finiteness growth, Geometric v. random walk boundaries, Stability, Cogrowth, Embedding left orderable groups, Schreier growth gap, $\ell^p$ models for hyperbolic groups, Characterizing hyperbolic groups <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.04207v1-abstract-full').style.display = 'none'; document.getElementById('2310.04207v1-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </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">10 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/2306.03846">arXiv:2306.03846</a> <span> [<a href="https://arxiv.org/pdf/2306.03846">pdf</a>, <a href="https://arxiv.org/ps/2306.03846">ps</a>, <a href="https://arxiv.org/format/2306.03846">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"> A realisation result for moduli spaces of group actions on the line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Brum%2C+J">Joaqu铆n Brum</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Rivas%2C+C">Crist贸bal Rivas</a>, <a href="/search/math?searchtype=author&query=Triestino%2C+M">Michele Triestino</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="2306.03846v4-abstract-short" style="display: inline;"> Given a finitely generated group $G$, the possible actions of $G$ on the real line (without global fixed points), considered up to semi-conjugacy, can be encoded by the space of orbits of a flow on a compact space $(Y, 桅)$ naturally associated with $G$ and uniquely defined up to flow equivalence, that we call the \emph{Deroin space} of $G$. We show a realisation result: every expansive flow… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.03846v4-abstract-full').style.display = 'inline'; document.getElementById('2306.03846v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2306.03846v4-abstract-full" style="display: none;"> Given a finitely generated group $G$, the possible actions of $G$ on the real line (without global fixed points), considered up to semi-conjugacy, can be encoded by the space of orbits of a flow on a compact space $(Y, 桅)$ naturally associated with $G$ and uniquely defined up to flow equivalence, that we call the \emph{Deroin space} of $G$. We show a realisation result: every expansive flow $(Y, 桅)$ on a compact metrisable space of topological dimension 1, satisfying some mild additional assumptions, arises as the Deroin space of a finitely generated group. This is proven by identifying the Deroin space of an explicit family of groups acting on suspension flows of subshifts, which is a variant of a construction introduced by the second and fourth authors. This result provides a source of examples of finitely generated groups satisfying various new phenomena for actions on the line, related to their rigidity/flexibility properties and to the structure of (path-)connected components of the space of actions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2306.03846v4-abstract-full').style.display = 'none'; document.getElementById('2306.03846v4-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2023. </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">32 pages; v4: minor corrections and references updated, final version to appear in the Journal of Topology</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.14545">arXiv:2305.14545</a> <span> [<a href="https://arxiv.org/pdf/2305.14545">pdf</a>, <a href="https://arxiv.org/format/2305.14545">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Liouville property for groups and conformal dimension </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Nekrashevych%2C+V">Volodymyr Nekrashevych</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+T">Tianyi Zheng</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="2305.14545v2-abstract-short" style="display: inline;"> Conformal dimension is a fundamental invariant of metric spaces, particularly suited to the study of self-similar spaces, such as spaces with an expanding self-covering (e.g. Julia sets of complex rational functions). The dynamics of these systems are encoded by the associated iterated monodromy groups, which are examples of contracting self-similar groups. Their amenability is a well-known open q… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.14545v2-abstract-full').style.display = 'inline'; document.getElementById('2305.14545v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.14545v2-abstract-full" style="display: none;"> Conformal dimension is a fundamental invariant of metric spaces, particularly suited to the study of self-similar spaces, such as spaces with an expanding self-covering (e.g. Julia sets of complex rational functions). The dynamics of these systems are encoded by the associated iterated monodromy groups, which are examples of contracting self-similar groups. Their amenability is a well-known open question. We show that if $G$ is an iterated monodromy group, and if the (Alfhors-regular) conformal dimension of the underlying space is strictly less than 2, then every symmetric random walk with finite second moment on $G$ has the Liouville property. As a corollary, every such group is amenable. This criterion applies to all examples of contracting groups previously known to be amenable, and to many new ones. In particular, it implies that for every post-critically finite complex rational function $f$ whose Julia set is not the whole sphere, the iterated monodromy group of $f$ is amenable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.14545v2-abstract-full').style.display = 'none'; document.getElementById('2305.14545v2-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 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </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, 5 figures, v2: minor changes</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.00091">arXiv:2209.00091</a> <span> [<a href="https://arxiv.org/pdf/2209.00091">pdf</a>, <a href="https://arxiv.org/format/2209.00091">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"> Solvable Groups and Affine Actions on the Line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Brum%2C+J">Joaqu铆n Brum</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Rivas%2C+C">Crist贸bal Rivas</a>, <a href="/search/math?searchtype=author&query=Triestino%2C+M">Michele Triestino</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="2209.00091v2-abstract-short" style="display: inline;"> We prove a structural result for orientation-preserving actions of finitely generated solvable groups on real intervals, considered up to semi-conjugacy. As applications we obtain new answers to a problem first considered by J. F. Plante, which asks under which conditions an action of a solvable group on a real interval is semi-conjugate to an action on the line by affine transformations. We show… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.00091v2-abstract-full').style.display = 'inline'; document.getElementById('2209.00091v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.00091v2-abstract-full" style="display: none;"> We prove a structural result for orientation-preserving actions of finitely generated solvable groups on real intervals, considered up to semi-conjugacy. As applications we obtain new answers to a problem first considered by J. F. Plante, which asks under which conditions an action of a solvable group on a real interval is semi-conjugate to an action on the line by affine transformations. We show that this is always the case for actions by $C^1$ diffeomorphisms on closed intervals. For arbitrary actions by homeomorphisms, for which this result is no longer true (as shown by Plante), we show that a semi-conjugacy to an affine action still exists in a local sense, at the level of germs near the endpoints. Finally for a vast class of solvable groups, including all solvable linear groups, we show that the family of affine actions on the line is robust, in the sense that any action by homeomorphisms on the line which is sufficiently close to an affine action must be semi-conjugate to an affine action. This robustness fails for general solvable groups, as illustrated by a counterexample. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.00091v2-abstract-full').style.display = 'none'; document.getElementById('2209.00091v2-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> 3 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 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">v2 incorporates suggestions of the referees (discussions of examples), and changes needed after the revision of the companion work arXiv:2104.14678; 39 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F16; 37C85; 20E08; 20F60; 57M60 (primary) 37E05; 37B05 (secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2206.06689">arXiv:2206.06689</a> <span> [<a href="https://arxiv.org/pdf/2206.06689">pdf</a>, <a href="https://arxiv.org/ps/2206.06689">ps</a>, <a href="https://arxiv.org/format/2206.06689">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"> Some torsion-free solvable groups with few subquotients </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boudec%2C+A+L">Adrien Le Boudec</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="2206.06689v3-abstract-short" style="display: inline;"> We construct finitely generated torsion-free solvable groups $G$ that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of $G$ are virtually abelian. In particular all finitely generated metabelian subgroups of $G$ are virtually abelian. The existence of such groups shows that there is no "torsion-free version" of P. Kropholler's theorem, which character… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.06689v3-abstract-full').style.display = 'inline'; document.getElementById('2206.06689v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2206.06689v3-abstract-full" style="display: none;"> We construct finitely generated torsion-free solvable groups $G$ that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of $G$ are virtually abelian. In particular all finitely generated metabelian subgroups of $G$ are virtually abelian. The existence of such groups shows that there is no "torsion-free version" of P. Kropholler's theorem, which characterises solvable groups of infinite rank via their metabelian subquotients. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.06689v3-abstract-full').style.display = 'none'; document.getElementById('2206.06689v3-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 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.11924">arXiv:2205.11924</a> <span> [<a href="https://arxiv.org/pdf/2205.11924">pdf</a>, <a href="https://arxiv.org/format/2205.11924">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"> Growth of actions of solvable groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boudec%2C+A+L">Adrien Le Boudec</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="2205.11924v2-abstract-short" style="display: inline;"> Given a finitely generated group $G$, we are interested in common geometric properties of all graphs of faithful actions of $G$. In this article we focus on their growth. We say that a group $G$ has a Schreier growth gap $f(n)$ if every faithful $G$-set $X$ satisfies $\mathrm{vol}_{G, X}(n)\succcurlyeq f(n)$, where $\mathrm{vol}_{G, X}(n)$ is the growth of the action of $G$ on $X$. Here we study S… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.11924v2-abstract-full').style.display = 'inline'; document.getElementById('2205.11924v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.11924v2-abstract-full" style="display: none;"> Given a finitely generated group $G$, we are interested in common geometric properties of all graphs of faithful actions of $G$. In this article we focus on their growth. We say that a group $G$ has a Schreier growth gap $f(n)$ if every faithful $G$-set $X$ satisfies $\mathrm{vol}_{G, X}(n)\succcurlyeq f(n)$, where $\mathrm{vol}_{G, X}(n)$ is the growth of the action of $G$ on $X$. Here we study Schreier growth gaps for finitely generated solvable groups. We prove that if a metabelian group $G$ is either finitely presented or torsion-free, then $G$ has a Schreier growth gap $n^2$, provided $G$ is not virtually abelian. We also prove that if $G$ is a metabelian group of Krull dimension $k$, then $G$ has a Schreier growth gap $n^k$. For instance the wreath product $C_p \wr \mathbb{Z}^d$ has a Schreier growth gap $n^d$, and $\mathbb{Z} \wr \mathbb{Z}^d$ has a Schreier growth gap $n^{d+1}$. These lower bounds are sharp. For solvable groups of finite Pr眉fer rank, we establish a Schreier growth gap $\exp(n)$, provided $G$ is not virtually nilpotent. This covers all solvable groups that are linear over $\mathbb{Q}$. Finally for a vast class of torsion-free solvable groups, which includes solvable groups that are linear, we establish a Schreier growth gap $n^2$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.11924v2-abstract-full').style.display = 'none'; document.getElementById('2205.11924v2-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 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">v2: updated version; Proposition 2.3 added</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.11099">arXiv:2203.11099</a> <span> [<a href="https://arxiv.org/pdf/2203.11099">pdf</a>, <a href="https://arxiv.org/ps/2203.11099">ps</a>, <a href="https://arxiv.org/format/2203.11099">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/s11856-024-2617-x">10.1007/s11856-024-2617-x <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A quantitative Neumann lemma for finitely generated groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gorokhovsky%2C+E">Elia Gorokhovsky</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Tamuz%2C+O">Omer Tamuz</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="2203.11099v2-abstract-short" style="display: inline;"> We study the coset covering function $\mathfrak{C}(r)$ of a finitely generated group: the number of cosets of infinite index subgroups needed to cover the ball of radius $r$. We show that $\mathfrak{C}(r)$ is of order at least $\sqrt{r}$ for all groups. Moreover, we show that $\mathfrak{C}(r)$ is linear for a class of amenable groups including virtually nilpotent and polycyclic groups, and that it… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.11099v2-abstract-full').style.display = 'inline'; document.getElementById('2203.11099v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.11099v2-abstract-full" style="display: none;"> We study the coset covering function $\mathfrak{C}(r)$ of a finitely generated group: the number of cosets of infinite index subgroups needed to cover the ball of radius $r$. We show that $\mathfrak{C}(r)$ is of order at least $\sqrt{r}$ for all groups. Moreover, we show that $\mathfrak{C}(r)$ is linear for a class of amenable groups including virtually nilpotent and polycyclic groups, and that it is exponential for property (T) groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.11099v2-abstract-full').style.display = 'none'; document.getElementById('2203.11099v2-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> 19 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">12 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Israel Journal of Mathematics, 2024 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.07765">arXiv:2107.07765</a> <span> [<a href="https://arxiv.org/pdf/2107.07765">pdf</a>, <a href="https://arxiv.org/ps/2107.07765">ps</a>, <a href="https://arxiv.org/format/2107.07765">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="Operator Algebras">math.OA</span> </div> </div> <p class="title is-5 mathjax"> Piecewise strongly proximal actions, free boundaries and the Neretin groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Caprace%2C+P">Pierre-Emmanuel Caprace</a>, <a href="/search/math?searchtype=author&query=Boudec%2C+A+L">Adrien Le Boudec</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="2107.07765v3-abstract-short" style="display: inline;"> A closed subgroup $H$ of a locally compact group $G$ is confined if the closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not contain the trivial subgroup. We establish a dynamical criterion on the action of a totally disconnected locally compact group $G$ on a compact space $X$ ensuring that no relatively amenable subgroup of $G$ can be confined. This property is equivalent… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.07765v3-abstract-full').style.display = 'inline'; document.getElementById('2107.07765v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.07765v3-abstract-full" style="display: none;"> A closed subgroup $H$ of a locally compact group $G$ is confined if the closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not contain the trivial subgroup. We establish a dynamical criterion on the action of a totally disconnected locally compact group $G$ on a compact space $X$ ensuring that no relatively amenable subgroup of $G$ can be confined. This property is equivalent to the fact that the action of $G$ on its Furstenberg boundary is free. Our criterion applies to the Neretin groups. We deduce that each Neretin group has two inequivalent irreducible unitary representations that are weakly equivalent. This implies that the Neretin groups are not of type I, thereby answering a question of Y.~Neretin. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.07765v3-abstract-full').style.display = 'none'; document.getElementById('2107.07765v3-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 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.14678">arXiv:2104.14678</a> <span> [<a href="https://arxiv.org/pdf/2104.14678">pdf</a>, <a href="https://arxiv.org/format/2104.14678">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"> Locally moving groups and laminar actions on the line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Brum%2C+J">Joaqu铆n Brum</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Rivas%2C+C">Crist贸bal Rivas</a>, <a href="/search/math?searchtype=author&query=Triestino%2C+M">Michele Triestino</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="2104.14678v3-abstract-short" style="display: inline;"> We prove various results that, given a sufficiently rich subgroup $G$ of the group of homeomorphisms on the real line, describe the structure of the other possible actions of $G$ on the line, and address under which conditions such actions must be semi-conjugate to the natural defining action of $G$. The main assumption is that $G$ should be locally moving, meaning that for every open interval the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.14678v3-abstract-full').style.display = 'inline'; document.getElementById('2104.14678v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.14678v3-abstract-full" style="display: none;"> We prove various results that, given a sufficiently rich subgroup $G$ of the group of homeomorphisms on the real line, describe the structure of the other possible actions of $G$ on the line, and address under which conditions such actions must be semi-conjugate to the natural defining action of $G$. The main assumption is that $G$ should be locally moving, meaning that for every open interval the subgroup of elements fixing pointwise its complement, acts on it without fixed points. We show that when $G$ is a locally moving group, every $C^1$ action of $G$ on the real line is semi-conjugate to its standard action or to a non-faithful action. The situation is much wilder when considering actions by homeomorphisms: for a large class of groups, we describe uncountably many conjugacy classes of faithful minimal actions. Next, we prove structure theorems for $C^0$ actions, based on the study of laminar actions, which are actions on the line preserving a lamination. When $G$ is a group of homeomorphisms of the line acting minimally, and with a non-trivial compactly supported element, then any faithful minimal action of $G$ on the line is either laminar or conjugate to its standard action. Moreover, when $G$ is a locally moving group with a suitable finite generation condition, for any faithful minimal laminar action there is a map from the lamination to the line, called a horograding, which is equivariant with respect to the action on the lamination and the standard one, and with some extra suitable conditions. This establishes a tight relation between all minimal actions on the line of such groups, and their standard actions. Finally, based on an analysis of the space of harmonic actions, we show that for a large class of locally moving groups, the standard action is locally rigid, in the sense that sufficiently small perturbations in the compact-open topology give semi-conjugate actions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.14678v3-abstract-full').style.display = 'none'; document.getElementById('2104.14678v3-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 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">205 pages, 11 figures; v2 is a major revision after report: title changed (previously 'Locally moving groups acting on the line and $\mathbb{R}$-focal actions'), structure reworked (chapters organized into 3 parts, each devoted to a single main theorem), many results strengthend to nearly optimal statements (requiring different approaches), digressions removed. To appear as an Ast茅risque volume</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 37C85. Secondary 20E08; 20F60; 37E05; 37B05; 57M60 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.03997">arXiv:2012.03997</a> <span> [<a href="https://arxiv.org/pdf/2012.03997">pdf</a>, <a href="https://arxiv.org/ps/2012.03997">ps</a>, <a href="https://arxiv.org/format/2012.03997">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"> Confined subgroups and high transitivity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boudec%2C+A+L">Adrien Le Boudec</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="2012.03997v2-abstract-short" style="display: inline;"> An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group $G$ admits a highly transitive action such that $G$ does not contain the subgroup of finitary alternating permutations, and if $H$ is a confined subgrou… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.03997v2-abstract-full').style.display = 'inline'; document.getElementById('2012.03997v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.03997v2-abstract-full" style="display: none;"> An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group $G$ admits a highly transitive action such that $G$ does not contain the subgroup of finitary alternating permutations, and if $H$ is a confined subgroup of $G$, then the action of $H$ remains highly transitive, possibly after discarding finitely many points. This result provides a tool to rule out the existence of highly transitive actions, and to classify highly transitive actions of a given group. We give concrete illustrations of these applications in the realm of groups of dynamical origin. In particular we obtain the first non-trivial classification of highly transitive actions of a finitely generated group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.03997v2-abstract-full').style.display = 'none'; document.getElementById('2012.03997v2-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> 19 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </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</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20B22; 20B35; 20B07; 37B05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.08677">arXiv:2006.08677</a> <span> [<a href="https://arxiv.org/pdf/2006.08677">pdf</a>, <a href="https://arxiv.org/ps/2006.08677">ps</a>, <a href="https://arxiv.org/format/2006.08677">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 commutator lemma for confined subgroups and applications to groups acting on rooted trees </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boudec%2C+A+L">Adrien Le Boudec</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="2006.08677v4-abstract-short" style="display: inline;"> A subgroup $H$ of a group $G$ is confined if the $G$-orbit of $H$ under conjugation is bounded away from the trivial subgroup in the space $\operatorname{Sub}(G)$ of subgroups of $G$. We prove a commutator lemma for confined subgroups. For groups of homeomorphisms, this provides the exact analogue for confined subgroups (hence in particular for URSs) of the classical commutator lemma for normal su… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.08677v4-abstract-full').style.display = 'inline'; document.getElementById('2006.08677v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.08677v4-abstract-full" style="display: none;"> A subgroup $H$ of a group $G$ is confined if the $G$-orbit of $H$ under conjugation is bounded away from the trivial subgroup in the space $\operatorname{Sub}(G)$ of subgroups of $G$. We prove a commutator lemma for confined subgroups. For groups of homeomorphisms, this provides the exact analogue for confined subgroups (hence in particular for URSs) of the classical commutator lemma for normal subgroups: if $G$ is a group of homeomorphisms of a Hausdorff space $X$ and $H$ is a confined subgroup of $G$, then $H$ contains the derived subgroup of the rigid stabilizer of some open subset of $X$. We apply this commutator lemma in the setting of groups acting on rooted trees. We prove a theorem describing the structure of URSs of weakly branch groups and of their non-topologically free minimal actions. Among the applications of these results, we show: 1) if $G$ is a finitely generated branch group, the $G$-action on $\partial T$ has the smallest possible growth among all faithful $G$-actions; 2) if $G$ is a finitely generated branch group, then every embedding from $G$ into a group of homeomorphisms of strongly bounded type (e.g. a bounded automaton group) must be spatially realized; 3) if $G$ is a finitely generated weakly branch group, then $G$ does not embed into the group IET of interval exchange transformations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.08677v4-abstract-full').style.display = 'none'; document.getElementById('2006.08677v4-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 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </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">49 pages, final version (v3->v4: typesetting fixed)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1906.05744">arXiv:1906.05744</a> <span> [<a href="https://arxiv.org/pdf/1906.05744">pdf</a>, <a href="https://arxiv.org/ps/1906.05744">ps</a>, <a href="https://arxiv.org/format/1906.05744">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/jlms.12521">10.1112/jlms.12521 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Triple transitivity and non-free actions in dimension one </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boudec%2C+A+L">Adrien Le Boudec</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="1906.05744v3-abstract-short" style="display: inline;"> The transitivity degree of a group $G$ is the supremum of all integers $k$ such that $G$ admits a faithful $k$-transitive action. Few obstructions are known to impose an upper bound on the transitivity degree for infinite groups. The results of this article provide two new classes of groups whose transitivity degree can be computed, as a corollary of a classification of all $3$-transitive actions… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.05744v3-abstract-full').style.display = 'inline'; document.getElementById('1906.05744v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.05744v3-abstract-full" style="display: none;"> The transitivity degree of a group $G$ is the supremum of all integers $k$ such that $G$ admits a faithful $k$-transitive action. Few obstructions are known to impose an upper bound on the transitivity degree for infinite groups. The results of this article provide two new classes of groups whose transitivity degree can be computed, as a corollary of a classification of all $3$-transitive actions of these groups. More precisely, suppose that $G$ is a subgroup of the homeomorphism group of the circle $\mathsf{Homeo}(\mathbb{S}^1)$ or the automorphism group of a tree $\mathsf{Aut}(\mathbb{T})$. Under natural assumptions on the stabilizers of the action of $G$ on $\mathbb{S}^1$ or $\partial \mathbb{T}$, we use the dynamics of this action to show that every faithful action of $G$ on a set that is at least $3$-transitive must be conjugate to the action of $G$ on one of its orbits in $\mathbb{S}^1$ or $\partial \mathbb{T}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.05744v3-abstract-full').style.display = 'none'; document.getElementById('1906.05744v3-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, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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">26 pages. v1-> v2: addition of an appendix and some minor corrections, v2->v3: Minor corrections and abstract rewritten. Final version to appear in the journal of the London Math. Soc</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.12256">arXiv:1811.12256</a> <span> [<a href="https://arxiv.org/pdf/1811.12256">pdf</a>, <a href="https://arxiv.org/ps/1811.12256">ps</a>, <a href="https://arxiv.org/format/1811.12256">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"> Groups of piecewise linear homeomorphisms of flows </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Triestino%2C+M">Michele Triestino</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="1811.12256v5-abstract-short" style="display: inline;"> To every dynamical system $(X,\varphi)$ over a totally disconnected compact space, we associate a left-orderable group $T(\varphi)$. It is defined as a group of homeomorphisms of the suspension of $(X,\varphi)$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple, and if… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.12256v5-abstract-full').style.display = 'inline'; document.getElementById('1811.12256v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.12256v5-abstract-full" style="display: none;"> To every dynamical system $(X,\varphi)$ over a totally disconnected compact space, we associate a left-orderable group $T(\varphi)$. It is defined as a group of homeomorphisms of the suspension of $(X,\varphi)$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple, and if it is a subshift then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system $(X,\varphi)$, the group $T(\varphi)$ does not have infinite subgroups with Kazhdan's property $(T)$. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally if $(X,\varphi)$ has a dense orbit then the isomorphism type of the group $T(\varphi)$ is a complete invariant of flow equivalence of the pair $\{\varphi, \varphi^{-1}\}$. In the appendix we describe a Polish group into which $T(\varphi)$ embeds densely. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.12256v5-abstract-full').style.display = 'none'; document.getElementById('1811.12256v5-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, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 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">27 pages, to appear in Compositio Mathematica</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 20F60; 20E32; 20F05. Secondary 37B10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.08567">arXiv:1803.08567</a> <span> [<a href="https://arxiv.org/pdf/1803.08567">pdf</a>, <a href="https://arxiv.org/format/1803.08567">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="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/10.1112/topo.12151">10.1112/topo.12151 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Property FW, differentiable structures, and smoothability of singular actions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lodha%2C+Y">Yash Lodha</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Triestino%2C+M">Michele Triestino</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="1803.08567v4-abstract-short" style="display: inline;"> We provide a smoothening criterion for group actions on manifolds by singular diffeomorphisms. We prove that if a countable group $螕$ has the fixed point property FW for walls (e.g. if it has property (T)), every aperiodic action of $螕$ by diffeomorphisms that are of class $C^r$ with countably many singularities is conjugate to an action by true diffeomorphisms of class $C^r$ on a homeomorphic (po… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.08567v4-abstract-full').style.display = 'inline'; document.getElementById('1803.08567v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.08567v4-abstract-full" style="display: none;"> We provide a smoothening criterion for group actions on manifolds by singular diffeomorphisms. We prove that if a countable group $螕$ has the fixed point property FW for walls (e.g. if it has property (T)), every aperiodic action of $螕$ by diffeomorphisms that are of class $C^r$ with countably many singularities is conjugate to an action by true diffeomorphisms of class $C^r$ on a homeomorphic (possibly non-diffeomorphic) manifold. As applications, we show that Navas's result for actions of Kazhdan groups on the circle, as well as the recent solutions to Zimmer's conjecture, generalise to aperiodic actions by diffeomorphisms with countably many singularities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.08567v4-abstract-full').style.display = 'none'; document.getElementById('1803.08567v4-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> 7 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">20 pages, to appear in Journal of Topology</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22D55; 37C85; 57M60 (Primary); 37E10 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1801.10133">arXiv:1801.10133</a> <span> [<a href="https://arxiv.org/pdf/1801.10133">pdf</a>, <a href="https://arxiv.org/ps/1801.10133">ps</a>, <a href="https://arxiv.org/format/1801.10133">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"> Rigidity properties of full groups of pseudogroups over the Cantor set </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="1801.10133v2-abstract-short" style="display: inline;"> We show that the (topological) full group of a minimal pseudogroup over the Cantor set satisfies various rigidity phenomena of topological dynamical and combinatorial nature. Our main result applies to its possible homomorphisms into other groups of homeomorphisms, and implies that arbitrary homomorphisms between the full groups of a vast class of pseudogroups must extend to continuous morphisms… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.10133v2-abstract-full').style.display = 'inline'; document.getElementById('1801.10133v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.10133v2-abstract-full" style="display: none;"> We show that the (topological) full group of a minimal pseudogroup over the Cantor set satisfies various rigidity phenomena of topological dynamical and combinatorial nature. Our main result applies to its possible homomorphisms into other groups of homeomorphisms, and implies that arbitrary homomorphisms between the full groups of a vast class of pseudogroups must extend to continuous morphisms between pseudogroups (in particular giving rise to equivariant maps at the level of spaces). As applications, we obtain explicit obstructions to the existence of embeddings between full groups in terms of invariants of the underlying pseudogroups (the geometry of their orbital graphs, the complexity function, dynamical homology), and provide a complete descriptions of all homomorphisms within various families of groups including the Higman-Thompson groups (more generally full groups of one sided shifts of finite type), full groups of minimal $\mathbb{Z}$-actions on the Cantor set, and a class of groups of interval exchanges. We next consider a combinatorial rigidity property of groups, which formalises the inability of a group to act on any set with Schreier graphs growing uniformly subexponentially, or more generally not faster than a given function $f(n)$. For the exponential function this is a well-known consequence of property $(T)$. We use full groups to provide a source of examples of groups which satisfy this property but satisfy a strong negation of property $(T)$. A key tool used in the proofs is the study of the dynamics of the conjugation action of groups on their space of subgroups, endowed with the Chabauty topology. In particular we classify the confined and the uniformly recurrent subgroups of full groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.10133v2-abstract-full').style.display = 'none'; document.getElementById('1801.10133v2-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> 10 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">70 pages. Completely rewritten version, with a different formulation of the main result, and several new results. The title has changed. Comments are 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/1709.06733">arXiv:1709.06733</a> <span> [<a href="https://arxiv.org/pdf/1709.06733">pdf</a>, <a href="https://arxiv.org/ps/1709.06733">ps</a>, <a href="https://arxiv.org/format/1709.06733">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"> Locally compact groups whose ergodic or minimal actions are all free </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boudec%2C+A+L">Adrien Le Boudec</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicolas Matte Bon</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="1709.06733v2-abstract-short" style="display: inline;"> We construct locally compact groups with no non-trivial Invariant Random Subgroups and no non-trivial Uniformly Recurrent Subgroups. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.06733v2-abstract-full" style="display: none;"> We construct locally compact groups with no non-trivial Invariant Random Subgroups and no non-trivial Uniformly Recurrent Subgroups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.06733v2-abstract-full').style.display = 'none'; document.getElementById('1709.06733v2-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 July, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.07101">arXiv:1702.07101</a> <span> [<a href="https://arxiv.org/pdf/1702.07101">pdf</a>, <a href="https://arxiv.org/ps/1702.07101">ps</a>, <a href="https://arxiv.org/format/1702.07101">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="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Realizing uniformly recurrent subgroups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Tsankov%2C+T">Todor Tsankov</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="1702.07101v2-abstract-short" style="display: inline;"> We show that every uniformly recurrent subgroup of a locally compact group is the family of stabilizers of a minimal action on a compact space. More generally, every closed invariant subset of the Chabauty space is the family of stabilizers of an action on a compact space on which the stabilizer map is continuous everywhere. This answers a question of Glasner and Weiss. We also introduce the notio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.07101v2-abstract-full').style.display = 'inline'; document.getElementById('1702.07101v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.07101v2-abstract-full" style="display: none;"> We show that every uniformly recurrent subgroup of a locally compact group is the family of stabilizers of a minimal action on a compact space. More generally, every closed invariant subset of the Chabauty space is the family of stabilizers of an action on a compact space on which the stabilizer map is continuous everywhere. This answers a question of Glasner and Weiss. We also introduce the notion of a universal minimal flow relative to a uniformly recurrent subgroup and prove its existence and uniqueness. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.07101v2-abstract-full').style.display = 'none'; document.getElementById('1702.07101v2-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 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 February, 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">v2: 10 pages, minor revision according to referee report</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.01651">arXiv:1605.01651</a> <span> [<a href="https://arxiv.org/pdf/1605.01651">pdf</a>, <a href="https://arxiv.org/ps/1605.01651">ps</a>, <a href="https://arxiv.org/format/1605.01651">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> </div> </div> <p class="title is-5 mathjax"> Subgroup dynamics and $C^\ast$-simplicity of groups of homeomorphisms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boudec%2C+A+L">Adrien Le Boudec</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="1605.01651v3-abstract-short" style="display: inline;"> We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise pr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.01651v3-abstract-full').style.display = 'inline'; document.getElementById('1605.01651v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.01651v3-abstract-full" style="display: none;"> We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple. We also prove the converse of a result of Haagerup and Olesen: if Thompson's group $F$ is non-amenable, then Thompson's group $T$ must be $C^\ast$-simple. Our results further provide sufficient conditions on a group of homeomorphisms under which uniformly recurrent subgroups can be completely classified. This applies to Thompson's groups $F$, $T$ and $V$, for which we also deduce rigidity results for their minimal actions on compact spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.01651v3-abstract-full').style.display = 'none'; document.getElementById('1605.01651v3-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 December, 2016; <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">45 pages. Minor changes</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1512.02133">arXiv:1512.02133</a> <span> [<a href="https://arxiv.org/pdf/1512.02133">pdf</a>, <a href="https://arxiv.org/ps/1512.02133">ps</a>, <a href="https://arxiv.org/format/1512.02133">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"> Full groups of bounded automaton groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="1512.02133v2-abstract-short" style="display: inline;"> We show that every bounded automaton group can be embedded in a finitely generated, simple amenable group. The proof is based on the study of the topological full groups associated to the Schreier dynamical system of the mother groups. We also show that if $\mathcal{G}$ is a minimal 茅tale groupoid with unit space the Cantor set, the group $[[\mathcal{G}]]_t$ generated by all torsion elements in th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1512.02133v2-abstract-full').style.display = 'inline'; document.getElementById('1512.02133v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1512.02133v2-abstract-full" style="display: none;"> We show that every bounded automaton group can be embedded in a finitely generated, simple amenable group. The proof is based on the study of the topological full groups associated to the Schreier dynamical system of the mother groups. We also show that if $\mathcal{G}$ is a minimal 茅tale groupoid with unit space the Cantor set, the group $[[\mathcal{G}]]_t$ generated by all torsion elements in the topological full group has simple commutator subgroup. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1512.02133v2-abstract-full').style.display = 'none'; document.getElementById('1512.02133v2-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 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 December, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 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">20 pages. v2: an assumption was removed from Theorem 2.7</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1503.04977">arXiv:1503.04977</a> <span> [<a href="https://arxiv.org/pdf/1503.04977">pdf</a>, <a href="https://arxiv.org/ps/1503.04977">ps</a>, <a href="https://arxiv.org/format/1503.04977">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/10.1017/etds.2016.32">10.1017/etds.2016.32 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Extensive amenability and an application to interval exchanges </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Juschenko%2C+K">Kate Juschenko</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Monod%2C+N">Nicolas Monod</a>, <a href="/search/math?searchtype=author&query=de+la+Salle%2C+M">Mikael de la Salle</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="1503.04977v1-abstract-short" style="display: inline;"> Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.04977v1-abstract-full').style.display = 'inline'; document.getElementById('1503.04977v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1503.04977v1-abstract-full" style="display: none;"> Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank~${\leq 2}$. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on IET and show that there are subgroups $G <IET$ admitting no finitely supported measure with trivial boundary. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1503.04977v1-abstract-full').style.display = 'none'; document.getElementById('1503.04977v1-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 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">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ergodic Theory Dynam. Systems. 38 (2018), 195-219 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1408.0762">arXiv:1408.0762</a> <span> [<a href="https://arxiv.org/pdf/1408.0762">pdf</a>, <a href="https://arxiv.org/format/1408.0762">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="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Topological full groups of minimal subshifts with subgroups of intermediate growth </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="1408.0762v1-abstract-short" style="display: inline;"> We show that every Grigorchuk group $G_蠅$ embeds in (the commutator subgroup of) the topological full group of a minimal subshift. In particular, the topological full group of a Cantor minimal system can have subgroups of intermediate growth, a question raised by Grigorchuk; it can also have finitely generated infinite torsion subgroups, as well as residually finite subgroups that are not elementa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.0762v1-abstract-full').style.display = 'inline'; document.getElementById('1408.0762v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1408.0762v1-abstract-full" style="display: none;"> We show that every Grigorchuk group $G_蠅$ embeds in (the commutator subgroup of) the topological full group of a minimal subshift. In particular, the topological full group of a Cantor minimal system can have subgroups of intermediate growth, a question raised by Grigorchuk; it can also have finitely generated infinite torsion subgroups, as well as residually finite subgroups that are not elementary amenable, answering questions of Cornulier. By estimating the word-complexity of this subshift, we deduce that every Grigorchuk group $G_蠅$ can be embedded in a finitely generated simple group that has trivial Poisson boundary for every simple random walk. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.0762v1-abstract-full').style.display = 'none'; document.getElementById('1408.0762v1-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 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">13 pages, 2 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1402.2234">arXiv:1402.2234</a> <span> [<a href="https://arxiv.org/pdf/1402.2234">pdf</a>, <a href="https://arxiv.org/ps/1402.2234">ps</a>, <a href="https://arxiv.org/format/1402.2234">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Subshifts with slow complexity and simple groups with the Liouville property </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</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="1402.2234v3-abstract-short" style="display: inline;"> We study random walk on topological full groups of subshifts, and show the existence of infinite, finitely generated, simple groups with the Liouville property. Results by Matui and Juschenko-Monod have shown that the derived subgroups of topological full groups of minimal subshifts provide the first examples of finitely generated, simple amenable groups. We show that if the (not necessarily minim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1402.2234v3-abstract-full').style.display = 'inline'; document.getElementById('1402.2234v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1402.2234v3-abstract-full" style="display: none;"> We study random walk on topological full groups of subshifts, and show the existence of infinite, finitely generated, simple groups with the Liouville property. Results by Matui and Juschenko-Monod have shown that the derived subgroups of topological full groups of minimal subshifts provide the first examples of finitely generated, simple amenable groups. We show that if the (not necessarily minimal) subshift has a complexity function that grows slowly enough (e.g. linearly), then every symmetric and finitely supported probability measure on the topological full group has trivial Poisson-Furstenberg boundary. We also get explicit upper bounds for the growth of F酶lner sets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1402.2234v3-abstract-full').style.display = 'none'; document.getElementById('1402.2234v3-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, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 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">18 pages, no figure. Revised version after referee report. Reorganized introduction, added some examples. To appear in GAFA</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1307.5652">arXiv:1307.5652</a> <span> [<a href="https://arxiv.org/pdf/1307.5652">pdf</a>, <a href="https://arxiv.org/format/1307.5652">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="Probability">math.PR</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/10.1214/15-AIHP697">10.1214/15-AIHP697 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Liouville property for groups acting on rooted trees </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Amir%2C+G">Gideon Amir</a>, <a href="/search/math?searchtype=author&query=Angel%2C+O">Omer Angel</a>, <a href="/search/math?searchtype=author&query=Bon%2C+N+M">Nicol谩s Matte Bon</a>, <a href="/search/math?searchtype=author&query=Vir%C3%A1g%2C+B">B谩lint Vir谩g</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="1307.5652v2-abstract-short" style="display: inline;"> We show that on groups generated by bounded activity automata, every symmetric, finitely supported probability measure has the Liouville property. More generally we show this for every group of automorphisms of bounded type of a rooted tree. For automaton groups, we also give a uniform upper bound for the entropy of convolutions of every symmetric, finitely supported measure. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1307.5652v2-abstract-full" style="display: none;"> We show that on groups generated by bounded activity automata, every symmetric, finitely supported probability measure has the Liouville property. More generally we show this for every group of automorphisms of bounded type of a rooted tree. For automaton groups, we also give a uniform upper bound for the entropy of convolutions of every symmetric, finitely supported measure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1307.5652v2-abstract-full').style.display = 'none'; document.getElementById('1307.5652v2-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 July, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 July, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2013. </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">Major changes in the statement and proof of Theorem 1, it now holds for all groups of automorphisms of bounded type, not necessarily finite-state. Final version, to appear in Annales de l'IHP</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Annales de l'Institut Henri Poincar茅, Probabilit茅s et Statistiques (Vol. 52, No. 4, pp. 1763-1783). 2016 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to 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