# Relative hyperbolicity for automorphisms of free products

## TL;DR

This paper establishes conditions under which automorphisms of free products lead to relatively hyperbolic semidirect products, extending previous results and applying to conjugacy problems in free products of abelian groups.

## Contribution

It proves that certain atoroidal automorphisms of free products produce relatively hyperbolic semidirect products, generalizing known theorems and applying to conjugacy problems.

## Key findings

- Semidirect products are relatively hyperbolic under specified automorphisms.
- Extension of Gautero-Lustig and Ghosh's theorem to free products.
- Application to conjugacy problem in free products of abelian groups.

## Abstract

We prove that for a free product $G$ with free factor system $\mathcal{G}$, any automorphism $\phi$ preserving $\mathcal{G}$, atoroidal (in a sense relative to $\mathcal{G}$) and none of whose power send two different conjugates of subgroups in $\mathcal{G}$ on conjugates of themselves by the same element, gives rise to a semidirect product $G\rtimes_\phi \mathbb{Z}$ that is relatively hyperbolic with respect to suspensions of groups in $\mathcal{G}$. We recover a theorem of Gautero-Lustig and Ghosh that, if $G$ is a free group, $\phi$ an automorphism of $G$, and $\mathcal{G}$ is its family of polynomially growing subgroups, then the semidirect product by $\phi$ is relatively hyperbolic with respect to the suspensions of these subgroups. We apply the first result to the conjugacy problem for certain automorphisms (atoroidal and toral) of free products of abelian groups.