Groups acting on graphs: Their automorphisms and their length functions

Abstract

Actions on frees are powerful tools for understanding the structure of a group and its automorphisms. In this thesis, we generalise several existing results in this field to larger classes of groups.

This is a three paper thesis; the main body of the work is contained in the following papers:
[1] Matthew Collins. Fixed points of irreducible, displacement one automorphisms of free products. Preprint, May2023, available at arXiv:2305.01451.
[2] Matthew Collins. Growth and displacement of free product automorphisms. Preprint, July 2023, available at arXiv:2307.13502.
[3] Matthew Collins and Armando Martino. Length functions on groups and actions on graphs. Preprint, July 2023, available atarXiv:2307.10760.

In [1], we prove that an irreducible, growth rate1 automorphism of a free product fixes a single point in outer space. This can be thought of as a generalisation of Dicks & Ventura’s classification of the irreducible, growth rate 1 automorphisms of free groups. It is well known for an irreducible free group automorphism that its growth rate is equal to the minimal Lipschitz displacement of its action on Culler-Vogtmann space. This follows as a consequence of the existence of train track representatives for the automorphism. In [2], we extend this result to the general - possibly reducible - case as well as to the free product situation where growth is replaced by ‘relative growth’. In [3], we study generalisations of Chiswell’s Theorem that 0-hyperbolic Lyndon length functions on groups always arise as based length functions of the group acting isometrically on a tree. We produce counter-examples to show that this Theorem fails if one replaces 0-hyperbolicitywith δ-hyperbolicity. We then propose a set of axioms for the length function on a finitely generated group that ensures the function is bi-Lipschitz equivalent to a (or any) length function of the group acting on its Cayley graph.

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