Commability and focal locally compact groups

TL;DR

This paper introduces the concept of commability for locally compact groups, classifies focal hyperbolic groups under this relation, and identifies a real parameter as a quasi-isometry invariant in the mixed case.

Contribution

It defines commability for locally compact groups and provides a classification of focal hyperbolic groups up to this relation, highlighting a key invariant.

Findings

Classification of focal hyperbolic locally compact groups up to commability

Identification of a real parameter as a quasi-isometry invariant in the mixed case

Establishment of commability as an equivalence relation involving cocompact inclusions and quotients

Abstract

We introduce the notion of commability between locally compact groups, namely the equivalence relation generated by cocompact inclusions and quotients by compact normal subgroups. We give a classification of focal hyperbolic locally compact groups up to commability. In the mixed case, it involves a real parameter, which is shown to be a quasi-isometry invariant.