The outer space of a free product
Copyright policy )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 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).
Updated reference. To appear in Proc. L.M.S
- Laboratoire de Mathématiques Nicolas Oresme France
- Université de Caen Normandie France
- INSA de Toulouse France
- Université de Toulouse France
- University of Caen Lower Normandy France
Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations, Topological methods in group theory, McCullough-Miller spaces, actions, Group Theory (math.GR), finitely generated groups, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], 510, torsion-free hyperbolic groups, Mathematics - Geometric Topology, free products, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], limit groups, FOS: Mathematics, contractible outer space, [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR], [MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT], 20F28, 20E06, Culler-Vogtmann spaces, 20E08, 20E08; 20E06; 20F28;20J06, Geometric Topology (math.GT), morphisms between trees, 20J06, Groups acting on trees, Automorphism groups of groups, Geometric group theory, virtual cohomological dimension, Mathematics - Group Theory, finite classifying spaces
Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations, Topological methods in group theory, McCullough-Miller spaces, actions, Group Theory (math.GR), finitely generated groups, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], 510, torsion-free hyperbolic groups, Mathematics - Geometric Topology, free products, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], limit groups, FOS: Mathematics, contractible outer space, [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR], [MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT], 20F28, 20E06, Culler-Vogtmann spaces, 20E08, 20E08; 20E06; 20F28;20J06, Geometric Topology (math.GT), morphisms between trees, 20J06, Groups acting on trees, Automorphism groups of groups, Geometric group theory, virtual cohomological dimension, Mathematics - Group Theory, finite classifying spaces
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