The abelianization of the congruence IA-automorphism group of a free group

Abstract

We consider the abelianizations of some normal subgroups of the automorphism group of a finitely generated free group. Let F n be a free group of rank n. For d 2, we consider a group consisting the automorphisms of F n which act trivially on the first homology group of F n with Z/dZ-coefficients. We call it the congruence IA-automorphism group of level d and denote it by I A n,d . Let I O n,d be the quotient group of the congruence IA-automorphism group of level d by the inner automorphism group of a free group. We determine the abelianization of I A n,d and I O n,d for n 2 and d 2. Furthermore, for n = 2 and odd prime p, we compute the integral homology groups of I A 2, p for any dimension.

The abelianization of the level d mapping class group

“…We determine especially H 1 (M g,r [d]; Z) for d = 2 and odd d 3, and H 1 (Γ g [d]; Z) for all d 2. This is analogous to results in [26,34] for the abelianizations of the level d congruence subgroups of Aut F n and GL(n; Z). To determine the abelianization H 1 (M g,r [2]; Z), we construct an injective homomorphism β σ : M g,1 [2] → Map(H 1 (Σ g ; Z 2 ), Z 8 ).…”

“…We denote by θ m : F n → H ⊗m ⊗ Z d the mth component of θ. Then, the map τ d is an Aut F n -equivariant homomorphism, as in [18, Lemmas 2C and 2D; 23, Theorem 3.1] (see also [34]).…”

On the abelianizations of congruence subgroups of Aut(F 2)

“…Our work is related to results of Satoh [11,12]. In his papers Satoh considers the kernel T n,m of the composition…”

“…For instance, in [5], F. Grunewald and A. Lubotzky use the groups Γ(G, π) to construct linear representations of the automorphism group Aut(F n ). Our work is related to results of T. Satoh [11,12], see also Section 1.2. The joint work [1] of E. Ribnere and the author can be seen as an accompanying paper to the present one.…”

About presentations of braid groups and their generalizations

“…The cohomology of P Σ n was computed by C. Jensen, J. McCammond, and J. Meier [44]. N. Kawazumi [47], T. Sakasai [68], T. Satoh [69] and A. Pettet [66] have given related cohomological information for IA n . The integral cohomology of the natural direct limit of the groups Aut(F n ) is given in work of S. Galatius [36].…”