On the automorphism group of generalized Baumslag–Solitar groups

Abstract

A generalized Baumslag–Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that $Out\left(G\right)$ either contains non-abelian free groups or is virtually nilpotent of class $\le$2. It has torsion only at finitely many primes.

One may decide algorithmically whether $Out\left(G\right)$ is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out$\left(G\right)$ virtually nilpotent.

If $G$ is unimodular (virtually $F_n\times \mathbb{Z}$), then $Out\left(G\right)$ is commensurable with a semi-direct product ${\mathbb{Z}}^k⋊Out\left(H\right)$ with $H$ virtually free.