Automorphisms, free products, and properties of groups through spaces of trees

Abstract

Given a group acting on a tree, one may obtain a space of such trees, which is acted on by automorphisms of the group. In this thesis, we study these actions to derive various results regarding automorphisms of free products (which act on particularly nice trees).

This is a three paper thesis; the main body of the work is contained in the following papers:
[1] Harry Iveson. A presentation for the group of pure symmetric outer automorphisms of a given splitting of a free product. Preprint, December 2024, available at arXiv:2412.04250.
[2] Harry Iveson. Generators for the pure symmetric outer automorphisms of a free product. Preprint, July 2025, available at arXiv:2507.16662.
[3] Harry Iveson, Armando Martino, Wagner Sgobbi, and Peter Wong with an appendix by Francesco Fournier-Facio. Property R∞ for groups with infinitely many ends. Preprint, April 2025, available at arXiv:2504.12002.

In [1] (Paper 1), we give a concise presentation for the group of pure symmetric outer automorphisms, Out_S(G), of a given splitting of a free product G=G1*...*Gn.
This is achieved by applying a theorem of K. S. Brown to a particular subcomplex, Cn, of Guirardel and Levitt’s Outer Space for a free product — a space of trees, viewed through the language of graphs of groups, via Bass–Serre theory. The application of Brown’s theorem occupies the first half of the paper, while the second half of the paper develops a rigorous argument as to why the subcomplex Cn is simply connected (a requirement of Brown’s theorem).

In [2] (Paper 2), we give a geometric proof that Out_S(G) is generated by Whitehead (and factor) automorphisms. The motivation behind this was to provide a direct proof for Proposition 3.1.3 (and hence also for Corollary 3.1.4) of Paper 1 [1]. That is, to show that a particular subcomplex (Cn, or in this case, its subcomplex Sn) of the spine of the Outer Space for a free product is path connected.

In [3] (Paper 3), we use actions on R-trees to show that any group with infinitely many ends (and in fact, a larger class of groups; those which act ‘sufficiently canonically’ on a ‘sufficiently nice’ R-tree) has the property R∞. That is, that every automorphism of the group has infinitely many twisted conjugacy classes. As a corollary, we see that the problem of if a group has property R∞ or not is undecidable amongst the class of finitely presented groups.

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Identifiers

ORCID for Harry Margaret John Iveson: orcid.org/0009-0009-3564-5873

ORCID for Armando Martino: orcid.org/0000-0002-5350-3029

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