Nielsen Realisation by Gluing: Limit Groups and Free Products

TL;DR

This paper extends classical theorems about free groups to free products, establishing fixed point results for automorphisms and proving that limit groups are CAT(0), with new methods involving Stallings' theorem.

Contribution

It generalizes Nielsen realisation and fixed point theorems from free groups to free products and limit groups, providing new proofs and insights.

Findings

Fixed point theorem for outer automorphisms on free splitting complex

Nielsen realisation for limit groups

Limit groups are CAT(0) spaces

Abstract

We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a free product of Guirardel--Levitt, as well as a relative version of the Nielsen realisation theorem, which in the case of free groups answers a question of Karen Vogtmann. We also prove Nielsen realisation for limit groups, and as a byproduct obtain a new proof that limit groups are CAT($0$). The proofs rely on a new version of Stallings' theorem on groups with at least two ends, in which some control over the behaviour of virtual free factors is gained.