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</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/2405.04817">arXiv:2405.04817</a> <span> [<a href="https://arxiv.org/pdf/2405.04817">pdf</a>, <a href="https://arxiv.org/format/2405.04817">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"> Visual right-angled Artin subgroups of two-dimensional right-angled Coxeter groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Edletzberger%2C+A">Alexandra Edletzberger</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="2405.04817v1-abstract-short" style="display: inline;"> There is a procedure, due to Dani and Levcovitz, for taking a finite simplicial graph (螕) and a subgraph (螞) of its complement, checking some conditions, and, if satisfied, producing a graph (螖) such that the right-angled Artin group with presentation graph (螖) is a finite index subgroup of the right-angled Coxeter group with presentation graph (螕). They do not tell us how to find (螞), given (螕).… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.04817v1-abstract-full').style.display = 'inline'; document.getElementById('2405.04817v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.04817v1-abstract-full" style="display: none;"> There is a procedure, due to Dani and Levcovitz, for taking a finite simplicial graph (螕) and a subgraph (螞) of its complement, checking some conditions, and, if satisfied, producing a graph (螖) such that the right-angled Artin group with presentation graph (螖) is a finite index subgroup of the right-angled Coxeter group with presentation graph (螕). They do not tell us how to find (螞), given (螕). We show, in the 2--dimensional case, that the existence of such a (螞) is connected to the graph property of satellite-dismantlabilty of (螕), and we use this to give an algorithm for producing a suitable (螞) or deciding that one does not exist. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.04817v1-abstract-full').style.display = 'none'; document.getElementById('2405.04817v1-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">11 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20F55 </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/2202.11626">arXiv:2202.11626</a> <span> [<a href="https://arxiv.org/pdf/2202.11626">pdf</a>, <a href="https://arxiv.org/format/2202.11626">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> </div> </div> <p class="title is-5 mathjax"> Asymptotic cones of snowflake groups and the strong shortcut property </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Hoda%2C+N">Nima Hoda</a>, <a href="/search/math?searchtype=author&query=Woodhouse%2C+D+J">Daniel J. Woodhouse</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="2202.11626v2-abstract-short" style="display: inline;"> We exhibit an infinite family of snowflake groups all of whose asymptotic cones are simply connected. Our groups have neither polynomial growth nor quadratic Dehn function, the two usual sources of this phenomenon. We further show that each of our groups has an asymptotic cone containing an isometrically embedded circle or, equivalently, has a Cayley graph that is not strongly shortcut. These are… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.11626v2-abstract-full').style.display = 'inline'; document.getElementById('2202.11626v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.11626v2-abstract-full" style="display: none;"> We exhibit an infinite family of snowflake groups all of whose asymptotic cones are simply connected. Our groups have neither polynomial growth nor quadratic Dehn function, the two usual sources of this phenomenon. We further show that each of our groups has an asymptotic cone containing an isometrically embedded circle or, equivalently, has a Cayley graph that is not strongly shortcut. These are the first examples of groups whose asymptotic cones contain `metrically nontrivial' loops but no topologically nontrivial ones. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.11626v2-abstract-full').style.display = 'none'; document.getElementById('2202.11626v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">67 pages, 13 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20F69; 51F30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.15923">arXiv:2006.15923</a> <span> [<a href="https://arxiv.org/pdf/2006.15923">pdf</a>, <a href="https://arxiv.org/format/2006.15923">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/10.1080/10586458.2021.1982079">10.1080/10586458.2021.1982079 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Short, highly imprimitive words yield hyperbolic one-relator groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Hoffmann%2C+C">Charlotte Hoffmann</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="2006.15923v1-abstract-short" style="display: inline;"> We give experimental support for a conjecture of Louder and Wilton saying that words of imprimitivity rank greater than two yield hyperbolic one-relator groups. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.15923v1-abstract-full" style="display: none;"> We give experimental support for a conjecture of Louder and Wilton saying that words of imprimitivity rank greater than two yield hyperbolic one-relator groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.15923v1-abstract-full').style.display = 'none'; document.getElementById('2006.15923v1-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 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">14 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20F67; 20F05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Experiment. Math. 32 (2023), no 4, 631-640 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.02119">arXiv:1810.02119</a> <span> [<a href="https://arxiv.org/pdf/1810.02119">pdf</a>, <a href="https://arxiv.org/format/1810.02119">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/s10711-019-00457-x">10.1007/s10711-019-00457-x <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Morse subsets of CAT(0) spaces are strongly contracting </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</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="1810.02119v1-abstract-short" style="display: inline;"> We prove that Morse subsets of CAT(0) spaces are strongly contracting. This generalizes and simplifies a result of Sultan, who proved it for Morse quasi-geodesics. Our proof goes through the recurrence characterization of Morse subsets. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.02119v1-abstract-full" style="display: none;"> We prove that Morse subsets of CAT(0) spaces are strongly contracting. This generalizes and simplifies a result of Sultan, who proved it for Morse quasi-geodesics. Our proof goes through the recurrence characterization of Morse subsets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.02119v1-abstract-full').style.display = 'none'; document.getElementById('1810.02119v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 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">3 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20F67 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Geom. Dedicata 204 (2020), no 1, 311--314 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1803.05782">arXiv:1803.05782</a> <span> [<a href="https://arxiv.org/pdf/1803.05782">pdf</a>, <a href="https://arxiv.org/format/1803.05782">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 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.2018.123">10.1017/etds.2018.123 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Cogrowth for group actions with strongly contracting elements </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arzhantseva%2C+G+N">Goulnara N. Arzhantseva</a>, <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</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="1803.05782v2-abstract-short" style="display: inline;"> Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$, and let $未_N$ and $未_G$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric then $未_N/未_G>1/2$. Our result appl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.05782v2-abstract-full').style.display = 'inline'; document.getElementById('1803.05782v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1803.05782v2-abstract-full" style="display: none;"> Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$, and let $未_N$ and $未_G$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric then $未_N/未_G>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk's original result on free groups with respect to a word metrics and a recent result of Jaerisch, Matsuzaki, and Yabuki on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1803.05782v2-abstract-full').style.display = 'none'; document.getElementById('1803.05782v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 July, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 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">9 pages, 3 figures; v2 11 pages, 3 figures adds some details, refactors proofs</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F69; 37C35; 20F65; 20F67; 20F36 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ergod. Th. Dynam. Sys. 40 (2020) 1738-1754 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1703.01482">arXiv:1703.01482</a> <span> [<a href="https://arxiv.org/pdf/1703.01482">pdf</a>, <a href="https://arxiv.org/format/1703.01482">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> <span class="tag is-small is-grey 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/10.1090/tran/7544">10.1090/tran/7544 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Metrizable Topology on the Contracting Boundary of a Group </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Mackay%2C+J+M">John M. Mackay</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="1703.01482v3-abstract-short" style="display: inline;"> The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1703.01482v3-abstract-full" style="display: none;"> The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.01482v3-abstract-full').style.display = 'none'; document.getElementById('1703.01482v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 February, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 March, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">v1: 26 pages, 3 figures; v2: 44 pages, 6 figures, additional results; v3: 46 pages, 7 figures, minor changes</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20F67 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Trans. Amer. Math. Soc. 372 (2019), no. 3, 1555-1600 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.01660">arXiv:1605.01660</a> <span> [<a href="https://arxiv.org/pdf/1605.01660">pdf</a>, <a href="https://arxiv.org/format/1605.01660">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</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/10.1515/agms-2016-0011">10.1515/agms-2016-0011 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quasi-isometries need not induce homeomorphisms of contracting boundaries with the Gromov product topology </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</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="1605.01660v2-abstract-short" style="display: inline;"> We consider a `contracting boundary' of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space. We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boun… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.01660v2-abstract-full').style.display = 'inline'; document.getElementById('1605.01660v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.01660v2-abstract-full" style="display: none;"> We consider a `contracting boundary' of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space. We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.01660v2-abstract-full').style.display = 'none'; document.getElementById('1605.01660v2-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 July, 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">5 pages, to appear in Analysis and Geometry in Metric Spaces</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Analysis and Geometry in Metric Spaces, 4 (2016), no. 1, 278-281 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1602.03767">arXiv:1602.03767</a> <span> [<a href="https://arxiv.org/pdf/1602.03767">pdf</a>, <a href="https://arxiv.org/format/1602.03767">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> </div> <div 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/10.4171/GGD/498">10.4171/GGD/498 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Negative curvature in graphical small cancellation groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arzhantseva%2C+G+N">Goulnara N. Arzhantseva</a>, <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Gruber%2C+D">Dominik Gruber</a>, <a href="/search/math?searchtype=author&query=Hume%2C+D">David Hume</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="1602.03767v3-abstract-short" style="display: inline;"> We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.03767v3-abstract-full').style.display = 'inline'; document.getElementById('1602.03767v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1602.03767v3-abstract-full" style="display: none;"> We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group $G$ containing an element $g$ that is strongly contracting with respect to one finite generating set of $G$ and not strongly contracting with respect to another. In the case of classical $C'(1/6)$ small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting. We show that many graphical $Gr'(1/6)$ small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups. In the course of our analysis we show that if the defining graph of a graphical $Gr'(1/6)$ small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.03767v3-abstract-full').style.display = 'none'; document.getElementById('1602.03767v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">40 pages, 14 figures, v2: improved introduction, updated statement of Theorem 4.4, v3: new title (previously: "Contracting geodesics in infinitely presented graphical small cancellation groups"), minor changes, to appear in Groups, Geometry, and Dynamics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F06 (Primary) 20F65; 20F67 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Groups Geom. Dyn. 13 (2019), no.2, 579-632 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1601.07147">arXiv:1601.07147</a> <span> [<a href="https://arxiv.org/pdf/1601.07147">pdf</a>, <a href="https://arxiv.org/format/1601.07147">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/S0305004116000530">10.1017/S0305004116000530 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quasi-isometries Between Groups with Two-Ended Splittings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Martin%2C+A">Alexandre Martin</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="1601.07147v1-abstract-short" style="display: inline;"> We construct `structure invariants' of a one-ended, finitely presented group that describe the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For groups satisfying two technical conditions, these invariants reduce the problem of quasi-isometry classification of such groups to the problem of relative quasi-isometry classification of the factors of their J… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.07147v1-abstract-full').style.display = 'inline'; document.getElementById('1601.07147v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1601.07147v1-abstract-full" style="display: none;"> We construct `structure invariants' of a one-ended, finitely presented group that describe the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For groups satisfying two technical conditions, these invariants reduce the problem of quasi-isometry classification of such groups to the problem of relative quasi-isometry classification of the factors of their JSJ decompositions. The first condition is that their JSJ decompositions have two-ended cylinder stabilizers. The second is that every factor in their JSJ decompositions is either `relatively rigid' or `hanging'. Hyperbolic groups always satisfy the first condition, and it is an open question whether they always satisfy the second. The same methods also produce invariants that reduce the problem of classification of one-ended hyperbolic groups up to homeomorphism of their Gromov boundaries to the problem of classification of the factors of their JSJ decompositions up to relative boundary homeomorphism type. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.07147v1-abstract-full').style.display = 'none'; document.getElementById('1601.07147v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">61pages, 6 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Math. Proc. Cambridge Philos. Soc. , 162 (2017), no. 2, 249-291 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1601.01897">arXiv:1601.01897</a> <span> [<a href="https://arxiv.org/pdf/1601.01897">pdf</a>, <a href="https://arxiv.org/format/1601.01897">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</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"> Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arzhantseva%2C+G+N">Goulnara N. Arzhantseva</a>, <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Gruber%2C+D">Dominik Gruber</a>, <a href="/search/math?searchtype=author&query=Hume%2C+D">David Hume</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="1601.01897v2-abstract-short" style="display: inline;"> We introduce and begin a systematic study of sublinearly contracting projections. We give two characterizations of Morse quasi-geodesics in an arbitrary geodesic metric space. One is that they are sublinearly contracting; the other is that they have completely superlinear divergence. We give a further characterization of sublinearly contracting projections in terms of projections of geodesic segme… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.01897v2-abstract-full').style.display = 'inline'; document.getElementById('1601.01897v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1601.01897v2-abstract-full" style="display: none;"> We introduce and begin a systematic study of sublinearly contracting projections. We give two characterizations of Morse quasi-geodesics in an arbitrary geodesic metric space. One is that they are sublinearly contracting; the other is that they have completely superlinear divergence. We give a further characterization of sublinearly contracting projections in terms of projections of geodesic segments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.01897v2-abstract-full').style.display = 'none'; document.getElementById('1601.01897v2-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 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">24 pages, 5 figures. v2: 22 pages, 5 figures. Correction in proof of Thm 7.1. Proof of Prop 4.2 revised for improved clarity. Other minor changes per referee comments. To appear in Documenta Mathematica</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F06; 20F65; 20F67 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1412.8582">arXiv:1412.8582</a> <span> [<a href="https://arxiv.org/pdf/1412.8582">pdf</a>, <a href="https://arxiv.org/format/1412.8582">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1515/jgth-2015-0038">10.1515/jgth-2015-0038 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Mapping tori of free group automorphisms, and the Bieri-Neumann-Strebel invariant of graphs of groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Levitt%2C+G">Gilbert Levitt</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="1412.8582v1-abstract-short" style="display: inline;"> Let $G$ be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from $G$ to $\mathbb{Z}$ have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing $G$ as the mapping torus of a free group automorphism. This is similar to the case for 3--manifold groups, and differen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.8582v1-abstract-full').style.display = 'inline'; document.getElementById('1412.8582v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1412.8582v1-abstract-full" style="display: none;"> Let $G$ be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from $G$ to $\mathbb{Z}$ have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing $G$ as the mapping torus of a free group automorphism. This is similar to the case for 3--manifold groups, and different from the case of mapping tori of exponentially growing free group automorphisms. The proof uses a hierarchical decomposition of $G$ and requires determining the Bieri-Neumann-Strebel invariant of the fundamental group of certain graphs of groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.8582v1-abstract-full').style.display = 'none'; document.getElementById('1412.8582v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 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">21 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20E; 57M </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Group Theory, 19 (2016), no. 2, 191-216 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1407.7834">arXiv:1407.7834</a> <span> [<a href="https://arxiv.org/pdf/1407.7834">pdf</a>, <a href="https://arxiv.org/format/1407.7834">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/S1461157015000108">10.1112/S1461157015000108 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Virtual Geometricity is Rare </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Manning%2C+J+F">Jason F. Manning</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1407.7834v3-abstract-short" style="display: inline;"> We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1407.7834v3-abstract-full" style="display: none;"> We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1407.7834v3-abstract-full').style.display = 'none'; document.getElementById('1407.7834v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 April, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 July, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages, 2 figures v2 adds a link to the computer scripts used in the paper; v3 13pages, to appear in LMS Journal of Computation and Mathematics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20E05; 20P05; 57M10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> LMS J. Comput. Math. 18 (2015) 444-455 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1404.4335">arXiv:1404.4335</a> <span> [<a href="https://arxiv.org/pdf/1404.4335">pdf</a>, <a href="https://arxiv.org/format/1404.4335">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/10.4171/GGD/364">10.4171/GGD/364 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Growth Tight Actions of Product Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Tao%2C+J">Jing Tao</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="1404.4335v2-abstract-short" style="display: inline;"> A group action on a metric space is called growth tight if the exponential growth rate of the group with respect to the induced pseudo-metric is strictly greater than that of its quotients. A prototypical example is the action of a free group on its Cayley graph with respect to a free generating set. More generally, with Arzhantseva we have shown that group actions with strongly contracting elemen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1404.4335v2-abstract-full').style.display = 'inline'; document.getElementById('1404.4335v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1404.4335v2-abstract-full" style="display: none;"> A group action on a metric space is called growth tight if the exponential growth rate of the group with respect to the induced pseudo-metric is strictly greater than that of its quotients. A prototypical example is the action of a free group on its Cayley graph with respect to a free generating set. More generally, with Arzhantseva we have shown that group actions with strongly contracting elements are growth tight. Examples of non-growth tight actions are product groups acting on the $L^1$ products of Cayley graphs of the factors. In this paper we consider actions of product groups on product spaces, where each factor group acts with a strongly contracting element on its respective factor space. We show that this action is growth tight with respect to the $L^p$ metric on the product space, for all $1<p\leq \infty$. In particular, the $L^\infty$ metric on a product of Cayley graphs corresponds to a word metric on the product group. This gives the first examples of groups that are growth tight with respect to an action on one of their Cayley graphs and non-growth tight with respect to an action on another, answering a question of Grigorchuk and de la Harpe. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1404.4335v2-abstract-full').style.display = 'none'; document.getElementById('1404.4335v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 February, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 April, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 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 v2 15 pages, minor changes, to appear in Groups, Geometry, and Dynamics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F67; 20F65; 37C35 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Groups, Geometry and Dynamics 10 (2016) no. 2, pp 753-770 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1403.2865">arXiv:1403.2865</a> <span>  </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"> Quasi-isometry Invariants from Decorated Trees of Cylinders of Two-Ended JSJ Decompositions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</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="1403.2865v2-abstract-short" style="display: inline;"> We construct quasi-isometry invariants of a one-ended finitely presented group by considering the tree of cylinders of a two-ended JSJ decomposition of the group. When the group satisfies additional quasi-isometric rigidity hypotheses we construct finer invariants by also considering relative amounts of stretching across edges of the tree of cylinders. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1403.2865v2-abstract-full" style="display: none;"> We construct quasi-isometry invariants of a one-ended finitely presented group by considering the tree of cylinders of a two-ended JSJ decomposition of the group. When the group satisfies additional quasi-isometric rigidity hypotheses we construct finer invariants by also considering relative amounts of stretching across edges of the tree of cylinders. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1403.2865v2-abstract-full').style.display = 'none'; document.getElementById('1403.2865v2-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> 27 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 March, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">arXiv:1601.07147 now contains all of these results and much, much more</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20E05; 20E06; 20E08; 20F65; 20F67 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1401.0499">arXiv:1401.0499</a> <span> [<a href="https://arxiv.org/pdf/1401.0499">pdf</a>, <a href="https://arxiv.org/format/1401.0499">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/pjm.2015.278.1">10.2140/pjm.2015.278.1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Growth Tight Actions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arzhantseva%2C+G+N">Goulnara N. Arzhantseva</a>, <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Tao%2C+J">Jing Tao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1401.0499v4-abstract-short" style="display: inline;"> We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting element and if $G$ is not too badly distorted in $\mathcal{X}$, then the action of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1401.0499v4-abstract-full').style.display = 'inline'; document.getElementById('1401.0499v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1401.0499v4-abstract-full" style="display: none;"> We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting element and if $G$ is not too badly distorted in $\mathcal{X}$, then the action of $G$ on $\mathcal{X}$ is a growth tight action. It follows that if $\mathcal{X}$ is a cocompact, relatively hyperbolic $G$--space, then the action of $G$ on $\mathcal{X}$ is a growth tight action. This generalizes all previously known results for growth tightness of cocompact actions: every already known example of a group that admits a growth tight action and has some infinite, infinite index normal subgroups is relatively hyperbolic, and, conversely, relatively hyperbolic groups admit growth tight actions. This also allows us to prove that many CAT(0) groups, including flip-graph-manifold groups and many Right Angled Artin Groups, and snowflake groups admit cocompact, growth tight actions. These provide first examples of non-relatively hyperbolic groups admitting interesting growth tight actions. Our main result applies as well to cusp uniform actions on hyperbolic spaces and to the action of the mapping class group on Teichmueller space with the Teichmueller metric. Towards the proof of our main result, we give equivalent characterizations of strongly contracting elements and produce new examples of group actions with strongly contracting elements. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1401.0499v4-abstract-full').style.display = 'none'; document.getElementById('1401.0499v4-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 March, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 January, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages, 4 figures v2 added references v3 40 pages, 6 figures, expanded preliminary sections to make paper more self-contained, other minor improvements v4 updated bibliography, to appear in Pacific Journal of Mathematics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F67; 20F65; 37C35; 20E06; 57Mxx </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Pacific J. Math. 278 (2015) 1-49 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1208.3652">arXiv:1208.3652</a> <span> [<a href="https://arxiv.org/pdf/1208.3652">pdf</a>, <a href="https://arxiv.org/format/1208.3652">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 Geometric Proof of the Structure Theorem for Cyclic Splittings of Free Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</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="1208.3652v2-abstract-short" style="display: inline;"> We give a geometric proof of a well known theorem that describes splittings of a free group as an amalgamated product or HNN extension over the integers. The argument generalizes to give a similar description of splittings of a virtually free group over a virtually cyclic group. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1208.3652v2-abstract-full" style="display: none;"> We give a geometric proof of a well known theorem that describes splittings of a free group as an amalgamated product or HNN extension over the integers. The argument generalizes to give a similar description of splittings of a virtually free group over a virtually cyclic group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.3652v2-abstract-full').style.display = 'none'; document.getElementById('1208.3652v2-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 August, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 August, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages, v2 added a reference</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20E05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Topology Proceedings 50 (2017), 335-349 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1009.2492">arXiv:1009.2492</a> <span> [<a href="https://arxiv.org/pdf/1009.2492">pdf</a>, <a href="https://arxiv.org/ps/1009.2492">ps</a>, <a href="https://arxiv.org/format/1009.2492">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/10.2140/agt.2016.16.621">10.2140/agt.2016.16.621 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Splitting Line Patterns in Free Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</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="1009.2492v5-abstract-short" style="display: inline;"> We construct a boundary of a finite rank free group relative to a finite list of conjugacy classes of maximal cyclic subgroups. From the cut points and uncrossed cut pairs of this boundary we construct a simplicial tree on which the group acts cocompactly. We show that the quotient graph of groups is the JSJ decomposition of the group relative to the given collection of conjugacy classes. This p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.2492v5-abstract-full').style.display = 'inline'; document.getElementById('1009.2492v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1009.2492v5-abstract-full" style="display: none;"> We construct a boundary of a finite rank free group relative to a finite list of conjugacy classes of maximal cyclic subgroups. From the cut points and uncrossed cut pairs of this boundary we construct a simplicial tree on which the group acts cocompactly. We show that the quotient graph of groups is the JSJ decomposition of the group relative to the given collection of conjugacy classes. This provides a characterization of virtually geometric multiwords: they are the multiwords that are built from geometric pieces. In particular, a multiword is virtually geometric if and only if the relative boundary is planar. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1009.2492v5-abstract-full').style.display = 'none'; document.getElementById('1009.2492v5-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 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 September, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages, 6 figures; v2 fixed a few typos; v3 38 pages, 21 figures; v4 30 pages, 11 figures 'Preliminaries' section expanded to make paper self-contained and split into two sections. Some arguments refactored and simplified. Paper streamlined; v5 56 pages, 21 figures Added examples and improved exposition according to referee comments. To appear in Algebraic & Geometric Topology</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20F67; 20E05; 57M05; 20E06 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Algebr. Geom. Topol. 16 (2016) 621-673 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1006.2123">arXiv:1006.2123</a> <span> [<a href="https://arxiv.org/pdf/1006.2123">pdf</a>, <a href="https://arxiv.org/format/1006.2123">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/10.2140/gt.2011.15.1419">10.2140/gt.2011.15.1419 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Line Patterns in Free Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</a>, <a href="/search/math?searchtype=author&query=Macura%2C+N">Natasa Macura</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="1006.2123v4-abstract-short" style="display: inline;"> We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a line pattern in a free group acts by isometries on a related space if and only if there are no cut pairs in the decomposition space. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1006.2123v4-abstract-full" style="display: none;"> We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a line pattern in a free group acts by isometries on a related space if and only if there are no cut pairs in the decomposition space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1006.2123v4-abstract-full').style.display = 'none'; document.getElementById('1006.2123v4-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 July, 2010; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 June, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages, 22 figures, PDFLatex; v2. finite index requires extra hypothesis; v3. 37 pages, 24 figures: updated references and add example in Section 6.3 of a rigid pattern for which the free group is not finite index in the group of pattern preserving quasi-isometries; v4. 40 pages, 26 figures: improved exposition and add example in Section 6.4 of a rigid pattern whose cube complex is not a tree</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20E05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Geom. Topol. 15 (2011) 1419-1475 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1001.0086">arXiv:1001.0086</a> <span> [<a href="https://arxiv.org/pdf/1001.0086">pdf</a>, <a href="https://arxiv.org/format/1001.0086">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"> Computing the Maximum Slope Invariant in Tubular Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H. Cashen</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="1001.0086v1-abstract-short" style="display: inline;"> We show that the maximum slope invariant for tubular groups is easy to calculate, and give an example of two tubular groups that are distinguishable by their maximum slopes but not by edge pattern considerations or isoperimetric function. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1001.0086v1-abstract-full" style="display: none;"> We show that the maximum slope invariant for tubular groups is easy to calculate, and give an example of two tubular groups that are distinguishable by their maximum slopes but not by edge pattern considerations or isoperimetric function. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1001.0086v1-abstract-full').style.display = 'none'; document.getElementById('1001.0086v1-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> 31 December, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, 7 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0707.1502">arXiv:0707.1502</a> <span> [<a href="https://arxiv.org/pdf/0707.1502">pdf</a>, <a href="https://arxiv.org/format/0707.1502">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4171/GGD/92">10.4171/GGD/92 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quasi-isometries Between Tubular Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cashen%2C+C+H">Christopher H Cashen</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="0707.1502v3-abstract-short" style="display: inline;"> We give a method of constructing maps between tubular groups inductively according to a set of strategies. This map will be a quasi-isometry exactly when the set of strategies is consistent. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for them. There is an algorithm that will in finite time either produce a consistent set of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0707.1502v3-abstract-full').style.display = 'inline'; document.getElementById('0707.1502v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0707.1502v3-abstract-full" style="display: none;"> We give a method of constructing maps between tubular groups inductively according to a set of strategies. This map will be a quasi-isometry exactly when the set of strategies is consistent. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for them. There is an algorithm that will in finite time either produce a consistent set of strategies or decide that such a set does not exist. Consequently, this algorithm decides whether or not the groups are quasi-isometric. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0707.1502v3-abstract-full').style.display = 'none'; document.getElementById('0707.1502v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 November, 2008; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 July, 2007; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2007. </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">44 pages, 11 figures. PDFLaTeX. Improved exposition and added some auxiliary material to make the paper more self contained, per referee's comments</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Groups Geom. Dyn. 4 (2010), no. 3, 473-516 </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 contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/license/index.html">Copyright</a></li> <li><a href="https://info.arxiv.org/help/policies/privacy_policy.html">Privacy Policy</a></li> </ul> </div> <div class="column sorry-app-links"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/web_accessibility.html">Web Accessibility Assistance</a></li> <li> <p class="help"> <a class="a11y-main-link" href="https://status.arxiv.org" target="_blank">arXiv Operational Status <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 256 512" class="icon filter-dark_grey" role="presentation"><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"/></svg></a><br> Get status notifications via <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/email/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg>email</a> or <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/slack/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-black" role="presentation"><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"/></svg>slack</a> </p> </li> </ul> </div> </div> </div>  </div> </footer> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/member_acknowledgement.js"></script> </body> </html>

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