On the Abelianization of Congruence Subgroups of Aut(F2)
Abstract
Let Fn be the free group of rank n and let Aut+(Fn) be its special automorphism group. For an epimorphism pi : Fn -> G of the free group Fn onto a finite group G we call Gamma+(G,pi) = f in Aut+(Fn) | pi*f = pi the standard congruence subgroup of Aut+(Fn) associated to G and pi. In the case n = 2 we fully describe the abelianization of Gamma+(G,pi) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Gamma+(G,pi) < Aut+(F2) has infinite abelianization.
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