The outer space of a free product

Vincent Guirardel, Gilbert Levitt

Published 2005-01-19, updated 2006-08-29Version 3

We associate a contractible ``outer space'' to any free product of groups G=G_1*...*G_q. It equals Culler-Vogtmann space when G is free, 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).