Deformation spaces of $G$-trees and automorphisms of Baumslag–Solitar groups

Abstract

We construct an invariant deformation retract of a deformation space of $G$-trees. We show that this complex is finite dimensional in certain cases and provide an example that is not finite dimensional. Using this complex we compute the automorphism group of the classical non-solvable Baumslag–Solitar groups BS($p,q$). The most interesting case is when $p$ properly divides $q$. Collins and Levin computed a presentation for Aut(BS($p,q$)) in this case using algebraic methods. Our computation uses Bass–Serre theory to derive these presentations. Additionally, we provide a geometric argument showing Out(BS($p,q$)) is not finitely generated when $p$ properly divides $q$.