Groups acting on hyperbolic spaces with a locally finite orbit

TL;DR

This paper investigates the structure of groups acting on hyperbolic spaces, especially quasitrees and their finite products, revealing strong algebraic constraints when such actions have locally finite orbits.

Contribution

It provides new insights into group actions on restricted hyperbolic spaces, establishing conditions under which the group structure is strongly constrained.

Findings

Strong conclusions on group structure with locally finite orbits

Analysis of group actions on finite products of quasitrees

Application to specific groups like Leary-Minasyan group

Abstract

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 what happens when a group acts isometrically on a restricted class of hyperbolic spaces, for instance quasitrees. We obtain strong conclusions on the group structure if the action has a locally finite orbit, especially if the group is finitely generated. We also look at group actions on finite products of quasitrees, where our actions may be by automorphisms or by isometries, including the Leary - Minasyan group.