ON THE ABELIANIZATION OF CONGRUENCE SUBGROUPS OF Aut(F2)

Let Fn be the free group of rank n and let Aut +(Fn) be its special automorphism group. For an epimorphism π : Fn → G of the free group Fn onto a finite group G we call Γ (G, π) = {φ ∈ Aut+(Fn) | πφ = π} the standard congruence subgroup of Aut+(Fn) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ(G, π) ≤ Aut+(F2) has infinite abelianization.