2015/06/19 by Amin Saied, Saied, Amin
Mathematics · #20F28 #20J06 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1506.05861
openalex publication_date 2015/06/19 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
The groups Γn,s are defined in terms of homotopy equivalences of certain graphs, and are natural generalisations of Out(Fn) and Aut(Fn). 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(Fn) is a virtual duality group in [1]. More recently, in [5], their cohomology Hi(Γn,s), over a field of characteristic zero, was computed in ranks n=1, 2 giving new constructions of unstable homology classes of Out(Fn) and Aut(Fn). In this paper we show that, for fixed i and n, this cohomology Hi(Γn,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 \Hi(Γn,s)\s≥0 satisfy representation stability, but with an improved stable range of s ≥ 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 \mathfrakSs-module Hi(Γn,s) for all s ≥ i+n. In particular this implies that, for fixed i and n, the dimension of Hi(Γn,s), is given by a polynomial in s for all s≥ i+n. We compute explicit examples of such character polynomials to demonstrate this phenomenon.