Groups acting faithfully on trees and properly on products of trees

TL;DR

This paper investigates which finitely generated groups can act properly on finite products of trees, providing evidence for hyperbolic surface groups and showing limitations for certain RAAGs and virtually special groups.

Contribution

It offers new insights into group actions on products of trees, especially regarding hyperbolic surface groups and restrictions for RAAGs and virtually special groups.

Findings

Hyperbolic surface groups can act properly on finite products of locally finite trees.

Many RAAGs do not admit proper actions on such products.

Some virtually special groups cannot act properly on any finite product of locally finite trees.

Abstract

We examine the question of which finitely generated groups act properly on a finite product of simplicial trees, considering both arbitrary trees and where all trees are locally finite. In the second case we present evidence in favour of hyperbolic surface groups having such an action. However we also present evidence that many RAAGs do not admit such an action and we give an example of a virtually special group which does not act properly preserving factors on any finite product of locally finite trees, even though it does so on a product of three trees without the local finiteness condition.