Related papers: Quasi-isometry rigidity of groups

We prove several rigidity properties for random quotients of mapping class groups of surfaces, namely whose kernel is normally generated by the n-th steps of finitely many independent random walks. Firstly, we generalise a celebrated…

Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there…

We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a…

Let $1 \to K \longrightarrow G \stackrel{\pi}\longrightarrow Q$ be an exact sequence of hyperbolic groups. Let $Q_1 < Q$ be a quasiconvex subgroup and let $G_1=\pi^{-1}(Q_1)$. Under relatively mild conditions (e.g. if $K$ is a closed…

We classify the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups with trivial or cyclic intersections.

A group with a geometric action on some hyperbolic space is necessarily word hyperbolic, but on the other hand every countable group acts (metrically) properly by isometries on a locally finite hyperbolic graph. In this paper we consider…

The aim of this paper is to present the main constructions of the substructures of an almost groupoid and to discuss their basic properties. The definitions and properties concerning these new algebraic constructions extend to almost…

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0)…

We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with…

We survey the known results regarding the boundaries of word-hyperbolic groups.

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is…

We show that finitely generated mapping tori of free groups have a canonical collection of maximal sub-mapping tori of finitely generated free groups with respect to which they are relatively hyperbolic and locally relatively quasi-convex.…

In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].

We describe structure of quasihomomorphisms from arbitrary groups to discrete groups. We show that all quasihomomorphisms are 'constructible', i.e., are obtained via certain natural operations from homomorphisms to some groups and…

We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many…

The goal of this paper is to construct quasi-isometrically embedded subgroups of Thompson's group $F$ which are isomorphic to $\fz^n$ for all $n$. A result estimating the norm of an element of Thompson's group is found. As a corollary,…

This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We…

This paper is a follow-up to our joint paper with I. Agol, P. Storm and K. Whyte "Finiteness of arithmetic hyperbolic reflection groups". The main purpose is to investigate the effective side of the method developed there and its possible…

We prove that, given a torsion-free relatively hyperbolic group G with non-relatively-hyperbolic peripherals, isomorphic finite index subgroups of G have the same index. This applies for instance to fundamental groups of finite-volume…

We study conditions under which quasi-conformal homeomorphisms are quasi-isometries. We show that if two nilpotent geodesic Lie groups are quasi-conformally homeomorphic, then they are quasi-isometrically equivalent. We also give more…