Automorphism growth and group decompositions

Abstract

Let $G$ be a finitely generated group with an automorphism $\phi \in \mathrm{Aut}(G)$, or an outer automorphism $\varphi \in \mathrm{Out}(G)$. Suppose that $G$ decomposes into simpler pieces on which the growth behaviour of $\phi$ and $\varphi$ 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 $\phi$ and $\varphi$ on the whole $G$.

15 pages; previously an appendix to arXiv:2501.12321, split off at the editor's request for separate publication