Nielsen Realization by Gluing: Limit Groups and Free Products

Abstract

We generalize the Karrass–Pietrowski–Solitar and the Nielsen realization 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 and Mosher and on the outer space of a free product of Guirardel and Levitt, and also a relative version of the Nielsen realization theorem, which, in the case of free groups, answers a question of Karen Vogtmann. We also prove Nielsen realization 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 behavior of virtual free factors is gained.