The outer space of a free product

We associate a contractible ‘outer space’ to any free product of groups G = G 1 * … * G q . It is identical to Culler–Vogtmann space when G is free, and McCullough–Miller space when no G i is ℤ. Our proof of contractibility (given when G is not free) is based on Skora's idea of deforming morphisms between trees. Using the action of Out( G ) on this space, we show that Out( G ) has finite virtual cohomological dimension, or is VFL (it has a finite index subgroup with a finite classifying space), if the groups G i and Out( G i ) have similar properties. We deduce that Out( G ) is VFL if G is a torsion‐free hyperbolic group, or a limit group (finitely generated fully residually free group).