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Proceedings of the London Mathematical Society Advance Access originally published online on February 8, 2007
Proceedings of the London Mathematical Society 2007 94(3):695-714; doi:10.1112/plms/pdl026
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© 2007 London Mathematical Society

The outer space of a free product

Vincent Guirardel1 and Gilbert Levitt2

1 Laboratoire Émile Picard
umr cnrs 5580
Université Paul Sabatier
31062 Toulouse cedex 4
France
guirardel{at}picard.ups-tlse.fr
2 LMNO
umr cnrs 6139
BP 5186
Université de Caen
14032 Caen cedex
France
levitt{at}math.unicaen.fr

Received 4 April 2005. Revision received 31 May 2006.

We associate a contractible ‘outer space’ to any free product of groups G = G1 * ... * Gq. It is identical to Culler–Vogtmann space when G is free, and McCullough–Miller space when no Gi 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 Gi and Out(Gi) 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).


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