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Homological stability for automorphism groups. (English) Zbl 1393.18006
The aim of the paper under review is to prove homology stability for several distinct families of groups \(G_n\), which include the following examples:
- symmetric groups \(\Sigma_n\) (Theorems D, E);
– braid groups \(\beta_n\) and surface braid groups \(\beta^S_n\) (Theorems D, F);
– automorphisms of free groups \(\mathrm{Aut}(F_n)\) (Theorem G);
– general linear groups \(\mathrm{GL}_n(R)\) over associative rings (Theorem 5.11);
– hyperbolic unitary groups \(\mathrm{U}^\epsilon_n(R, \Lambda)\) over form rings (Theorem H);
– mapping class groups of surfaces (Theorem I);
– mapping class groups of 3-manifolds (Theorem J).
The property shared by the above families of groups is that they all can be realized as automorphism groups of objects \(A\oplus X^{\oplus n}, n \in \mathbb{N}\) for some choice of objects \(A, X \in Ob(\mathcal{G})\) in some braided monoidal groupoid \((\mathcal{G}, \oplus, 0)\).
A key feature of the present paper is that the homology stability problem is solved not only for trivial coefficients but also for abelian coefficient modules and twisted polynomial coefficient systems (see Sections 3 and 4.4, respectively). For example, sign representations of symmetric groups \(\Sigma_n\) can be made into an abelian coefficient module, while Burau representations of braid groups are an example of a polynomial coefficient system of degree 1.
The language of braided monoidal groupoids and closely related pre-braided homogeneous categories is used to both define the twisted coefficient systems and formulate the general Theorems 3.1, 3.4, 4.20 reducing the original question of homology stability to proving that certain semisimplicial sets \(W_n(A, X)\), called spaces of destabilisations, are highly connected.
This categorical approach allows the authors to treat the problem of homology stability with constant and twisted coefficients simultaneously, and thus, leads to a number of new results on stability with twisted coefficients for the families of groups mentioned above.

MSC:
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
19B14 Stability for linear groups
20G10 Cohomology theory for linear algebraic groups
20E05 Free nonabelian groups
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
20G35 Linear algebraic groups over adèles and other rings and schemes
57M99 General low-dimensional topology
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