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

摘要

Let F _n be the free group of rank n, and let Aut ~+(F _n) be its special automorphism group. For an epimorphism π: F _n → G of the free group F _n onto a finite group G we call Γ + (G, π)={φ ∈ Aut ~+ (F _n) {pipe} πφ=π} the standard congruence subgroup of Aut ~+(F _n) 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 ~+(F 2) has infinite abelianization.