Nielsen Realisation by Gluing: Limit Groups and Free Products
  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. 
Sebastian Hensel and Dawid Kielak
math.GR math.GT