Incommensurable lattices in Baumslag-Solitar complexes

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.

v2: 34 pages. The notion of depth profile has been modified slightly to exclude elliptic elements. There is some new exposition with additional references