Relative Nielsen realisation for free products

Sebastian Hensel, Dawid Kielak

Published 2016-01-10Version 1

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, as well as a relative version of the Nielsen realisation theorem, which is new even for free groups. 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.