Groups acting on trees with almost prescribed local action

TL;DR

This paper explores groups acting on regular trees with specified local actions, presenting new examples of simple groups with finite asymptotic dimension, and analyzing their properties including actions on CAT(0) cube complexes.

Contribution

It introduces novel constructions of simple groups with finite asymptotic dimension and examines their lattice structures and actions on geometric complexes.

Findings

Existence of finitely generated simple groups of asymptotic dimension one.

Examples of simple groups with simple lattices.

Construction of groups acting on trees with prescribed local actions.

Abstract

We investigate a family of groups acting on a regular tree, defined by prescribing the local action almost everywhere. We study lattices in these groups and give examples of compactly generated simple groups of finite asymptotic dimension (actually one) not containing lattices. We also obtain examples of simple groups with simple lattices, and we prove the existence of (infinitely many) finitely generated simple groups of asymptotic dimension one. We also prove various properties of these groups, including the existence of a proper action on a CAT(0) cube complex.