Local topological properties of asymptotic cones of groups

Conner, Gregory R.; Kent, Curtis

doi:10.2140/agt.2014.14.1413

Cited by 2 publications

(4 citation statements)

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“…Moreover, Gromov's conjectural dichotomy was proved to hold in several cases (see e.g. [56,14] for results in this direction).…”

Section: Asymptotic Conesmentioning

confidence: 98%

Quasi-isometric rigidity of piecewise geometric manifolds

Frigerio¹

2016

Preprint

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Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are always quasi-isometric, and a group Γ is quasi-isometrically rigid if every group quasi-isometric to Γ is virtually isomorphic to Γ. In this survey we describe quasi-isometric rigidity results for fundamental groups of manifolds which can be decomposed into geometric pieces. After stating by now classical results on lattices in semisimple Lie groups, we focus on the class of fundamental groups of 3-manifolds, and describe the behaviour of quasi-isometries with respect to the Milnor-Kneser prime decomposition (following Papasoglu and Whyte) and with respect to the JSJ decomposition (following Kapovich and Leeb). We also discuss quasi-isometric rigidity results for fundamental groups of higher dimensional graph manifolds, that were recently defined by Lafont, Sisto and the author. Our main tools are the study of geometric group actions and quasi-actions on Riemannian manifolds and on trees of spaces, via the analysis of the induced actions on asymptotic cones.

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“…Moreover, Gromov's conjectural dichotomy was proved to hold in several cases (see e.g. [56,14] for results in this direction).…”

Section: Asymptotic Conesmentioning

confidence: 98%

Quasi-isometric rigidity of piecewise geometric manifolds

Frigerio¹

2016

Preprint

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“…Remark 3.3. In [4], the author with Greg Conner note that such groups are uniformly locally simply connected; specifically, every loop of length r bounds a disc of diameter at most Kr where K only depends on the group. However, the discs are not necessarily Lipschitz.…”

Section: Wide Groups and Ends Of Asymptotic Conesmentioning

confidence: 99%

“…In particular, he asked whether the following dichotomy is true: the fundamental group of an asymptotic cone of a finitely generated group is always either trivial or of order continuum. One reason for this question was that asymptotic cones of nilpotent groups are simply connected (Pansu,[19]), the same is true for hyperbolic groups since all cones in that case are R-trees, but asymptotic cones of many solvable non-nilpotent groups (say, the Baumslag-Solitar group BS(2, 1) or Sol) contain Hawaiian earrings, and that seems to be a common property of very many other groups [2], [4].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic cones of HNN extensions and amalgamated products

Kent

2014

Algebr. Geom. Topol.

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Gromov asked whether an asymptotic cone of a finitely generated group was always simply connected or had uncountable fundamental group. We prove that Gromov's dichotomy holds for asympotic cones with cut points, as well as, HNN extensions and amalgamated products where the associated subgroups are nicely embedded. We also show a slightly weaker dichotomy for multiple HNN extensions of free groups.

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Local topological properties of asymptotic cones of groups

Cited by 2 publications

References 23 publications

Quasi-isometric rigidity of piecewise geometric manifolds

Quasi-isometric rigidity of piecewise geometric manifolds

Asymptotic cones of HNN extensions and amalgamated products

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