Homology stability for outer automorphism groups of free 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 F I of finite sets and injections relevant to the present context, and define polynomiality for functors in the context of pre-braided monoidal categories.A family of groupsis said to satisfy homological stability if the induced mapsare isomorphisms in a range 0 ≤ i ≤ f (n) increasing with n. In this paper, we prove that homological stability always holds if there is a monoidal category C satisfying a certain hypothesis and a pair of objects A and X in C, such that G n is the group of automorphisms of A ⊕ X ⊕n in C. We show that stability holds not just for constant coefficients, but also for both polynomial and abelian coefficients, without any further assumption on C. The polynomial coefficient systems considered here are functors F : C → Z -Mod satisfying a finite degree condition. They are generalisations of polynomial functors in the sense of functor homology, classically considered in homological stability, and include new examples such as the Burau representation of braid groups. Abelian coefficients are given by functors F : C → ZG ab ∞ -Mod for G ab ∞ the abelianisation of the limit group G ∞ , satisfying the same finite degree condition. These include coefficients such as the sign representation, or determinant-twisted polynomial functors. Such coefficient systems are newer to the subject. One consequence of stability with abelian coefficients is that stability with polynomial coefficients also holds (under the same conditions) for the commutator subgroups G n ≤ G n .Our theorem applies to all the classical examples and gives new stability results with twisted coefficients in particular for symmetric groups, alternating groups, unitary groups, braid groups, mapping class groups of non-orientable surfaces, automorphisms and symmetric automorphisms of free groups, and it proves stability for these groups...

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“…Alternatively, we could have shown that S n (e, Z) is isomorphic to the complex V ± n,1 of [45] and use Proposition 2 of that paper.…”

Section: 2mentioning

confidence: 99%

Homological stability for automorphism groups

Randal-Williams

Wahl

2017

Advances in Mathematics

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“…For the symmetric group this was proved by Nakaoka ([Nak60]) and for Aut(F n ) by Hatcher and Vogtmann ([HV98a], [HV04]). The homology groups H k (Σ n ) are completely known.…”

mentioning

confidence: 88%

“…The groups Aut(F n ) are special cases of a more general series of groups A s n , studied in [HV04] and [HVW06]. We recall the definition.…”

Section: Corollary 12 the Groupsmentioning

confidence: 99%

Stable homology of automorphism groups of free groups

Galatius

2011

Ann. Math.

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“…Hatcher and Vogtmann [37] (see also Hatcher [35]) proved that the homology of Out F n stabilizes in a certain stable range. This is an analogue of Harer's stability theorem [34] for the mapping class group.…”

Section: Outer Automorphism Group Of Free Groupsmentioning

confidence: 98%

Cohomological structure of the mapping class group and beyond

Morita¹

2006

Proceedings of Symposia in Pure Mathematics

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Abstract. In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces.Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders.

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Homology stability for outer automorphism groups of free groups

Cited by 57 publications

References 10 publications

Homological stability for automorphism groups

Homological stability for automorphism groups

Stable homology of automorphism groups of free groups

Cohomological structure of the mapping class group and beyond

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