On the quasi-isometric classification of locally compact groups

Abstract

This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous negatively curved manifolds. A main conjecture provides a general description; an extended discussion reduces this conjecture to more specific statements.In the course of the paper, we provide statements of quasi-isometric rigidity for general symmetric spaces of noncompact type and also discuss accessibility issues in the realm of compactly generated locally compact groups.

Quasiisometries of negatively curved homogeneous manifolds associated with Heisenberg groups

“…The quasiisometric classification question asks for a classification of all negatively curved solvable Lie groups up to quasiisometry. The quasiisometric classification question for negatively curved solvable Lie groups is part of the larger project of quasiisometric classification of focal hyperbolic groups [3]. In this context, Dymarz [5] recently obtained results similar to our Theorems 1.1 and 1.3 for mixed-type focal hyperbolic groups.…”

Groups acting on trees with almost prescribed local action

“…Since the fundamental group of a Cayley graph of G is generated by loops of bounded length because G is compactly presented, the group G must be quasi-isometric to a tree according to [FW07, Theorem 1.1]. This implies that the group G must act geometrically on some locally finite tree (see [Cor12,Theorem 4.A.1] and references therein), and since G is unimodular, it follows from Lemma 5.3 that every locally elliptic open subgroup of G must be compact. In particular vertex stabilizers in G for its action on T d are compact, so the first statement is proved.…”

Non-amenability and visual Gromov hyperbolic spaces

“…as ρ(u, v) ≥ δ/c(µ) by (i) which gives the the right-hand side of (5). To prove the left-hand side of (5) let u, v ∈ ΓH(Z) (0) be two vertices.…”

“…Acknowledgements I would like to thank Ilkka Holopainen for his advice and for providing me with unpublished notes written by Aleksi Vähäkangas in 2007 on global Sobolev inequalities on Gromov hyperbolic spaces; Jussi Väisälä for providing me with the letters of correspondence between him and Oded Schramm from the end of 2004 with regard to the paper Embeddings of Gromov hyperbolic spaces, and to whom Lemma 1 is attributed; Piotr Nowak for many enjoyable discussions on growth homology; Yves de Cornulier for bringing his quasi-survey [5] to my attention; and Pekka Pankka for several suggestions on how to improve the text. I would also like to thank the Technion for its hospitality during my stay from January to May 2014, Uri Bader and Tobias Hartnick for many stimulating conversations, and Eline Zehavi for all her help during this stay.…”