Asymptotic cones of HNN extensions and amalgamated products

Kent, Curt

doi:10.2140/agt.2014.14.551

Cited by 3 publications

(2 citation statements)

References 29 publications

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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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“…Kent [29] defines a prairie group to be one for which every asymptotic cone is simply connected. As far as we are aware, the only groups currently known to be prairie groups are either virtually nilpotent or have quadratic Dehn function or are built by combining such pieces: direct products of prairie groups are prairie groups, and groups that are hyperbolic relative to prairie groups are prairie groups [16].…”

Section: Asymptotic Conesmentioning

confidence: 99%

Asymptotic cones of snowflake groups and the strong shortcut property

Cashen¹,

Hoda²,

Woodhouse³

2022

Preprint

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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.

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Densely related groups

Cornulier¹,

Boudec²

2019

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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We study the class of densely related groups. These are finitely generated (or more generally, compactly generated locally compact) groups satisfying a strong negation of being finitely presented, in the sense that new relations appear at all scales. Here, new relations means relations that do not follow from relations of smaller size. Being densely related is a quasiisometry invariant among finitely generated groups.We check that a densely related group has none of its asymptotic cones simply connected. In particular a lacunary hyperbolic group cannot be densely related.We prove that the Grigorchuk group is densely related. We also show that a finitely generated group that is (infinite locally finite)-by-cyclic and which satisfies a law must be densely related. Given a class C of finitely generated groups, we consider the following dichotomy: every group in C is either finitely presented or densely related. We show that this holds within the class of nilpotent-by-cyclic groups and the class of metabelian groups. In contrast, this dichotomy is no longer true for the class of 3-step solvable groups.

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Asymptotic cones of HNN extensions and amalgamated products

Cited by 3 publications

References 29 publications

Quasi-isometric rigidity of piecewise geometric manifolds

Quasi-isometric rigidity of piecewise geometric manifolds

Asymptotic cones of snowflake groups and the strong shortcut property

Densely related groups

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