Simple groups without lattices

TL;DR

This paper demonstrates that the group of almost automorphisms of a d-regular tree is an example of a simple, locally compact group that does not contain any lattices, a novel finding in the field.

Contribution

It provides the first known example of a simple, compactly generated, locally compact group without lattices.

Findings

The group of almost automorphisms of a d-regular tree admits no lattices.

This is the first example among simple locally compact groups.

The result impacts understanding of lattice existence in simple groups.

Abstract

We show that the group of almost automorphisms of a d-regular tree does not admit lattices. As far as we know this is the first such example among (compactly generated) simple locally compact groups.