Barking Up the Right Tree: Group Actions on Trees, Automorphisms and Separability

Actions on trees are powerful tools for understanding the structure of a group. In this thesis, we use them to understand separability and automorphisms of free products, and automorphisms of free-by-cyclic groups. This is a three paper thesis; the main body of the work is contained in the following papers: [1] Naomi Andrew. A Bass–Serre theoretic proof of a theorem of Burns and Romanovskii. Preprint, July 2021, available at arXiv:2107.02548. [2] Naomi Andrew. Serre’s property (FA) for automorphism groups of free products. J. Group Theory, 24(2):385–414, 2021. [3] Naomi Andrew and Armando Martino. Free-by-cyclic groups, automorphisms and actions on nearly canonical trees. Preprint, June 2021, available at arXiv:2106.02541. In [1], we use properties of actions on trees – described in the combinatorial language of graphs of groups, due to Bass and Serre – to re-prove that free products of subgroup separable groups are themselves subgroup separable. In [2], we suppose G is a free product of groups and investigate when Aut(G) admits actions on trees. Under the assumption that the factor groups are freely indecomposable and not Z (for example, if they are finite) this depends only on a count of the isomorphism classes appearing in the decomposition. To build actions on trees we use length functions and a theorem of Culler and Morgan; to rule them out we use commutation relations within the automorphism groups. In [3], we investigate the outer automorphisms of free-by-cyclic groups, and in some cases prove that they are finitely generated. To do this we introduce the notion of a “nearly canonical” action on a tree, construct such an action for certain free-by-cyclic groups, and use these actions to understand the outer automorphisms