On the FI-module structure of $H^i(Γ_{n,s})$

The groups $Γ_{n,s}$ are defined in terms of homotopy equivalences of certain graphs, and are natural generalisations of $\mbox{Out}(F_n)$ and $\mbox{Aut}(F_n)$. They have appeared frequently in the study of free group automorphisms, for example in proofs of homological stability in [8,9] and in the proof that Out$(F_n)$ is a virtual duality group in [1]. More recently, in [5], their cohomology $H^i(Γ_{n,s})$, over a field of characteristic zero, was computed in ranks $n=1, 2$ giving new constructions of unstable homology classes of $\mbox{Out}(F_n)$ and $\mbox{Aut}(F_n)$. In this paper we show that, for fixed $i$ and $n$, this cohomology $H^i(Γ_{n,s})$ forms a finitely generated FI-module of stability degree $n$ and weight $i$, as defined by Church-Ellenberg-Farb in [2]. We thus recover that for all $i$ and $n$, the sequences $\{H^i(Γ_{n,s})\}_{s\geq0}$ satisfy representation stability, but with an improved stable range of $s \geq i+n$ which agrees with the low dimensional calculations made in [5]. Another important consequence of this FI-module structure is the existence of character polynomials which determine the character of the $\mathfrak{S}_s$-module $H^i(Γ_{n,s})$ for all $s \geq i+n$. In particular this implies that, for fixed $i$ and $n$, the dimension of $H^i(Γ_{n,s})$, is given by a polynomial in $s$ for all $s\geq i+n$. We compute explicit examples of such character polynomials to demonstrate this phenomenon.

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