Automorphism growth and group decompositions

TL;DR

This paper investigates how the growth rates of automorphisms and outer automorphisms of finitely generated groups can be deduced from their behavior on simpler decomposed parts of the group, such as direct products or free products.

Contribution

It develops methods to infer the overall growth behavior of automorphisms from their known effects on decomposed components of the group.

Findings

Provides a framework for analyzing automorphism growth via group decompositions

Establishes connections between local and global growth behaviors

Offers tools for understanding automorphisms in complex group structures

Abstract

Let $G$ be a finitely generated group with an automorphism $\varphi\in{\rm Aut}(G)$, or an outer automorphism $\phi\in{\rm Out}(G)$. Suppose that $G$ decomposes into simpler pieces on which the growth behaviour of $\varphi$ and $\phi$ is known, particularly as a direct product, free product, or graph of groups. This article is devoted to the (often not entirely straightforward) problem of deducing information about the growth rates of $\varphi$ and $\phi$ on the whole $G$.