The Abelianization of the Congruence Ia-automorphism Group of a Free Group

Let Fn be a free group of rank n. An automorphism of Fn is called an IA-automorphism if it trivially acts on the abelianization H of Fn. We denote by IAn the group of IA-automorphisms and call it the IA-automorphism group of Fn. For any integer d ≥ 2, let IAn,d be the group of automorphisms of Fn which trivially acts on H⊗ZZ/dZ. We call IAn,d the congruence IA-automorphism group of Fn of level d. In this paper we determine the abelianization of IAn,d for n ≥ 2 and d ≥ 2. Furthermore, for any odd prime integer p, we give some remarks on the (co)homology groups of IAn,p with trivial coefficients. In particullar, we show that the second cohomology group of IAn,p has non-trivial p-torsion elements for n ≥ 9 and, we completely calculate the homology groups of IA2,p for any dimension.