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The outer space of a free product
Author(s) -
Guirardel Vincent,
Levitt Gilbert
Publication year - 2007
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdl026
Subject(s) - outer space , product (mathematics) , space (punctuation) , business , computer science , geometry , mathematics , operating system
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).