Incommensurable lattices in Baumslag-Solitar complexes

TL;DR

This paper investigates the structure of Baumslag-Solitar complexes, showing that their automorphism groups contain incommensurable uniform lattices that are finitely presented, torsion-free, and have isomorphic Cayley graphs.

Contribution

It demonstrates the existence of incommensurable uniform lattices within automorphism groups of Baumslag-Solitar complexes, with specific properties like finite presentation and torsion-freeness.

Findings

Existence of incommensurable uniform lattices in Aut($X_{m,n}$)

Lattices are finitely presented and torsion-free

Lattices admit isomorphic Cayley graphs

Abstract

This paper concerns locally finite 2-complexes $X_{m,n}$ which are combinatorial models for the Baumslag-Solitar groups $BS(m,n)$. We show that, in many cases, the locally compact group Aut($X_{m,n}$) contains incommensurable uniform lattices. The lattices we construct also admit isomorphic Cayley graphs and are finitely presented, torsion-free, and coherent.