A non-residually finite group acting uniformly properly on a hyperbolic space

Abstract

In this article we produce an example of a non-residually finite group which admits a uniformly proper action on a Gromov hyperbolic space. * The first author acknowledges the support of the ANR grant DAGGER ANR-16-CE40-0006-01. He is also grateful to the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment.

Finitely generated groups acting uniformly properly on hyperbolic space

“…The action of H W on Z W preserves C W and acts freely on this subgraph. The following theorem now follows from Proposition 3.3 of [4].…”

“…For instance: acylindrically hyperbolic groups [13] or relatively hyperbolic groups [6]. In this paper we will study the class of groups with uniformly proper actions studied in [4]. Definition 1.1.…”

“…The class of groups acting uniformly properly on hyperbolic spaces includes all subgroups of hyperbolic groups. In [4], the question of whether these two classes coincide is asked. This is shown to be false in [8], where uncountably many groups of type F P 2 acting uniformly properly on hyperbolic spaces are obtained, while there are only countably many finitely generated subgroups of hyperbolic groups.…”

Groups acting on hyperbolic spaces with a locally finite orbit

“…We finish this section by noting some examples in the literature of groups possessing a uniformly proper action on some hyperbolic space. This class is considered in [10] where it is denoted by P. They also consider the subclass P 0 of all groups with a uniformly proper action on a bounded valence hyperbolic graph, which is the same as the groups in Proposition 8.3. This in turn contains the class S of all subgroups of word hyperbolic groups.…”

Finitely generated groups acting uniformly properly on hyperbolic space

“…There have been many interesting generalisations of this notion, for instance: acylindrically hyperbolic groups [14] or relatively hyperbolic groups [6]. In this paper, we will study the class of groups with uniformly proper actions studied in [4]. Definition 1.1.…”