Assembling homology classes in automorphism groups of free groups

Abstract

The observation that a graph of rank n can be assembled from graphs of smaller rank k with s leaves by pairing the leaves together leads to a process for assembling homology classes for Out(Fn) and Aut(Fn) from classes for groups Γ k,s , where the Γ k,s generalize Out(F k ) = Γ k,0 and Aut(F k ) = Γ k,1 . The symmetric group Ss acts on H * (Γ k,s ) by permuting leaves, and for trivial rational coefficients we compute the Ss-module structure on H * (Γ k,s ) completely for k ≤ 2. Assembling these classes then produces all the known nontrivial rational homology classes for Aut(Fn) and Out(Fn) with the possible exception of classes for n = 7 recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of Aut(Fn) and Out(Fn) which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.

The topology and geometry of automorphism groups of free groups

“…In M * n,s one may collapse any or all of the subgraphs in a system without collapsing the whole graph, thereby coning off (and killing) the entire assembled class. This observation gives some (admittedly weak) credence to the conjecture made in [30] that all of the homology of M n,s below dimension 2n+s−3 is in the image of assembly maps.…”

Moduli spaces of colored graphs

“…For a precise definition and further reading on these groups with applications in geometric group theory we refer to [CHKV16] and the survey in [Vog16].…”

“…The purpose of this article is to define and study moduli spaces of colored graphs in the spirit of [HV98] and [CHKV16] where such spaces for uncolored graphs were used to study the homology of automorphism groups of free groups. Our motivation stems from the connection between these constructions in geometric group theory and the study of Feynman integrals as pointed out in [BK15].…”

“…In fact, the cases s = 0 and s = 1 encode the group homology of Out(F n ) and Aut(F n ), an active area of research. For the case s > 1 the rational homology of MG n,s is fully determined for n = 1, 2 in[CHKV16]. For n = 1 we have (4.1) H k (MG 1,s ; Q) = Q ( s−1 k ) if k ≤ s − 1 is even 0 otherwise.…”

The $\mathsf {Lie}$ Lie algebra

“…Let V ∧n be the Σ n -representation which is V ⊗n as a vector space, with Σ n acting with the sign of the permutation. We have the following theorem [10,7] Theorem 1.5. There is an isomorphism…”