# Serre's Property (FA) for automorphism groups of free products

## TL;DR

This paper characterizes when automorphism groups of free products of certain groups have Property (FA), extending previous results and fully resolving the finite cyclic case.

## Contribution

It provides necessary and sufficient conditions for Property (FA) in automorphism groups of free products, generalizing and completing prior work.

## Key findings

- Finite groups satisfy the conditions for Property (FA)
- Conditions are necessary and sufficient for free products of certain groups
- Resolves the open case for finite cyclic groups

## Abstract

We provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore this paper generalises the work of Leder (arXiv:1810.06287) for finite cyclic groups, as well as resolving the open case of that paper.