LATTICES IN BAUMSLAG-SOLITAR COMPLEXES

Abstract

This dissertation classifies the pairs of nonzero integers (m,n) for which the locally compact group of combinatorial automorphisms, Aut(X_{m,n}), contains incommensurable torsion-free lattices, where X_{m,n} is the combinatorial model for the Baumslag-Solitar group BS(m,n). In particular, we show that Aut(X_{m,n}) contains abstractly incommensurable torsion-free lattices if and only if there exists a prime p less than or equal to gcd(m,n) such that either m/gcd(m,n) or n/gcd(m,n) is divisible by p. In all these cases, we construct infinitely many commensurability classes. Additionally, we show that when Aut(X_{m,n}) does not contain incommensurable lattices, the cell complex X_{m,n} satisfies Leighton’s property. We further describe the structure of uniform lattices in the combinatorial automorphism group of a generalized Baumslag-Solitar complex and exhibit an example of a uniform lattice in Aut(X_{8,16}) that is not virtually torsion-free.