On the quasi-isometric classification of locally compact groups

TL;DR

This paper surveys the quasi-isometric classification of locally compact groups, focusing on amenable hyperbolic groups and negatively curved manifolds, proposing a main conjecture and discussing rigidity and accessibility issues.

Contribution

It introduces a main conjecture for classifying locally compact groups up to quasi-isometry and discusses rigidity results and accessibility in this context.

Findings

Quasi-isometric rigidity for symmetric spaces of noncompact type

Reduction of the main conjecture to specific statements

Discussion of accessibility in compactly generated 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.