Realising all countable groups as quasi-isometry groups

TL;DR

This paper constructs uncountably many proper geodesic metric spaces with a prescribed countable group as their quasi-isometry group, extending to hyperbolic groups and utilizing strong quasi-isometric rigidity phenomena.

Contribution

It provides a method to realize any countable group as the quasi-isometry group of a constructed metric space, including hyperbolic groups, using new strongly quasi-isometrically rigid spaces.

Findings

Uncountably many quasi-isometry classes constructed for any countable group.

Spaces constructed are hyperbolic if the group is hyperbolic.

Utilizes strong quasi-isometric rigidity of symmetric spaces.

Abstract

Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are hyperbolic metric spaces. We make use of a rigidity phenomenon for quasi-isometries exhibited by many symmetric spaces, called strong quasi-isometric rigidity. Our method involves the construction of new examples of strongly quasi-isometrically rigid spaces, arising as graphs of strongly quasi-isometrically rigid rank-one symmetric spaces.