Commability of groups quasi-isometric to trees

TL;DR

This paper investigates the concept of commability among locally compact groups acting on trees, establishing that such groups are all commable and providing bounds on the number of homomorphisms needed for commability.

Contribution

It proves all non-elementary groups acting on locally finite trees are commable, answers a question of Cornulier, and introduces the minimal number of homomorphisms needed for commability.

Findings

All non-elementary groups acting on locally finite trees are commable.

Six homomorphisms always suffice for commability.

First example of commable groups without fewer than six homomorphisms.

Abstract

Commability is the finest equivalence relation between locally compact groups such that $G$ and $H$ are equivalent whenever there is a continuous proper homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier, we show that all non-elementary locally compact groups acting geometrically on locally finite simplicial trees are commable, thereby strengthening previous forms of quasi-isometric rigidity for trees. We further show that 6 homomorphisms always suffice, and provide the first example of a pair of locally compact groups which are commable but without commation consisting of less than 6 homomorphisms. Our strong quasi-isometric rigidity also applies to products of symmetric spaces and Euclidean buildings, possibly with some factors being trees.