Solvable Baumslag-Solitar Lattices

TL;DR

This paper classifies lattices in the isometry groups of canonical models of solvable Baumslag-Solitar groups, revealing they lack strong rigidity but satisfy a weaker automorphism-based rigidity.

Contribution

It provides a complete classification of lattices in the isometry groups of Baumslag-Solitar model spaces and analyzes their rigidity properties.

Findings

Lattices in these groups are not strongly rigid.

There exist automorphisms of lattices not extending to the ambient group.

Lattices are related through automorphisms of the ambient group, satisfying a weaker rigidity.

Abstract

The solvable Baumslag Solitar groups $\text{BS}(1,n)$ each admit a canonical model space, $X_n$. We give a complete classification of lattices in $G_n = \text{Isom}^+(X_n)$ and find that such lattices fail to be strongly rigid$\unicode{x2014}$there are automorphisms of lattices $\Gamma \subset G_n$ which do not extend to $G_n$$\unicode{x2014}$but do satisfy a weaker form of rigidity: for all isomorphic lattices $\Gamma_1,\Gamma_2\subset G_n$, there is an automorphism $\rho \in \text{Aut}(G_n)$ so that $\rho(\Gamma_1) = \Gamma_2$.