Homology stability for outer automorphism groups of free groups

Abstract

We prove that the quotient map from Aut(F_n) to Out(F_n) induces an isomorphism on homology in dimension i for n at least 2i+4. This corrects an earlier proof by the first author and significantly improves the stability range. In the course of the proof, we also prove homology stability for a sequence of groups which are natural analogs of mapping class groups of surfaces with punctures. In particular, this leads to a slight improvement on the known stability range for Aut(F_n), showing that its i-th homology is independent of n for n at least 2i+2.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-54.abs.html

References in corpus (3)

• Stabilization for the automorphisms of free groups with boundaries
• Automorphisms of free groups with boundaries
• Morita classes in the homology of automorphism groups of free groups

Cited by in corpus (31)

• Stabilization for mapping class groups of 3-manifolds
• Representation stability and finite linear groups
• Homological stability for moduli spaces of high dimensional manifolds. I
• Stabilization for the automorphisms of free groups with boundaries
• Assembling homology classes in automorphism groups of free groups
• Erratum to: Homology stability for outer automorphism groups of free groups
• Homological stability for automorphism groups
• Tethers and homology stability for surfaces
• Homological stability for families of Coxeter groups
• The homology of the Brauer algebras
• On the edge of the stable range
• Hodge decomposition for stable homology of automorphism groups of free groups
• Erratum to ``Stabilization for the automorphisms of free groups with boundaries''
• Morita classes in the homology of automorphism groups of free groups
• The stable cohomology of automorphisms of free groups with coefficients in the homology representation
• The homology of the Temperley-Lieb algebras
• Higher operations in string topology of classifying spaces
• The homology of a Temperley-Lieb algebra on an odd number of strands
• Morita classes in the homology of Aut(F_n) vanish after one stabilization
• Stability phenomena in the topology of moduli spaces
• Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs
• Commensurations of subgroups of $\mathrm{Out}(F_N)$
• Direct products of free groups in $\mathrm{Aut}(F_N)$
• Mapping class subgroups of Out(F_n)
• A Quillen Stability Criterion for Bounded Cohomology
• Abelian covers of graphs and maps between outer automorphism groups of free groups
• Buildings, Group Homology and Lattices
• Stable cohomology of $Aut(F_n)$ with bivariant twisted coefficients
• Normal Tori in ${♯}_n(S^2\times S^1)$
• Topological Field Theories and Harrison Homology
• Secondary characteristic classes for subgroups of automorphism groups of free groups