On automorphism groups of free products of finite groups, I: Proper Actions

A generating set for the automorphism group of a graph product of abelian groups

We find a set of generators for the automorphism group of a graph product of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup of star-automorphisms defined in [7]. We follow closely the plan of M. Laurence's paper [11].

Diagonal complexes and the integral homology of the automorphism group of a free product

The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product. We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of diagonal complexes is introduced. This offers a generalisation of the theory of right-angled Artin groups in that each diagonal complex defines what we call a diagonal right-angled Artin group (DRAAG).

CTs for free products

The fundamental group of a finite graph of groups with trivial edge groups is a free product. We are interested in those outer automorphisms of such a free product that permute the conjugacy classes of the vertex groups. We show that in particular cases of interest, such as where vertex groups are themselves finite free products of finite and cyclic groups, given such an outer automorphism, after passing to a positive power, the outer automorphism is represented by a particularly nice kind of relative train track map called a CT. CTs were first introduced by Feighn and Handel for outer automor…

The Tits alternative for the automorphism group of a free product

Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all $i\in\{1,\dots,k\}$, both $G_i$ and its outer automorphism group $\text{Out}(G_i)$ satisfy the Tits alternative, then $\text{Out}(G)$ satisfies the Tits alternative. As an application, we prove that the Tits alternative holds for outer automorphism groups of right-angled Artin groups, and of torsion-free groups that are hyperbolic relative to a finite family of virtuall…

Boundaries of relative factor graphs and subgroup classification for automorphisms of free products

Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy classes of each $G_i$. We show that every finitely generated subgroup $H\subseteq Out(G,\mathcal{F})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ …

自同构增长与群分解

Automorphism growth and group decompositions

本文研究通过群分解推断自同构和外自同构在整体群上的增长速率问题。