Fixed subgroups of generalised Baumslag-Solitar groups

Abstract

We investigate fixed subgroups of automorphisms of generalised Baumslag-Solitar (GBS) groups. Our main results are for automorphisms leaving a Bass-Serre tree invariant, under the assumption that all edge stabilisers are strictly contained in the corresponding vertex stabilisers. We completely characterise which GBS groups admit such an automorphism with a fixed subgroup which is not finitely-generated. In doing so, we provide an infinite family of examples of non-finitely generated fixed subgroups in GBS groups. Dropping the above assumptions, we show that all finite order automorphisms of GBS groups have finitely generated fixed subgroups. Furthermore, we show that when the GBS graph is a tree, all automorphisms have finitely generated fixed subgroups.