Homological stability for automorphism groups

Abstract

Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial and abelian twisted coefficients, with no further assumptions. This new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class groups. By making systematic the proofs of classical stability results, we show that they all hold with the same type of coefficient systems, obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups, braid groups, automorphisms of free groups, unitary groups, mapping class groups of non-orientable surfaces and mapping class groups of 3-manifolds. Our construction can also be applied to families of groups not considered before in the context of homological stability. As a byproduct of our work, we construct the braided analogue of the category FI of finite sets and injections relevant to the present context, and define polynomiality for functors in the context of pre-braided monoidal categories.

v2: Major revision. Stability with abelian coefficient systems added as well as new examples. Added coauthor. v3: minor revision. v4: Final version, to appear in Advances in Math

References in corpus (8)

• Representation stability and finite linear groups
• Representations of categories of G-maps
• Noetherian property of infinite EI categories
• The homology of the Higman-Thompson groups
• Homological stability for automorphism groups of RAAGs
• The stable cohomology of automorphisms of free groups with coefficients in the homology representation
• Homology of braid groups, the Burau representation, and ${\mathbb{F}}_q$-points on superelliptic curves
• The first homology group of the mapping class group of a nonorientable surface with twisted coefficients

Cited by in corpus (16)

• Representation stability and finite linear groups
• FI-modules and the cohomology of modular representations of symmetric groups
• Representations of categories of G-maps
• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• Central stability for the homology of congruence subgroups and the second homology of Torelli groups
• Extension between functors from groups
• Homological stability for families of Coxeter groups
• On the edge of the stable range
• Homological stability for automorphism groups of RAAGs
• Homological vanishing for the Steinberg representation
• Generalized Long-Moody functors
• Representation stability for filtrations of Torelli groups
• Homology of braid groups, the Burau representation, and ${\mathbb{F}}_q$-points on superelliptic curves
• A comparison of twisted coefficient systems
• Representation stability for sequences of 0-Hecke modules
• Homology of the braid group with coefficients in the reduced Burau representation