On the Index of Congruence Subgroups of Aut(Fn)
Abstract
For an epimorphism pi of the free group Fn onto a finite group G write Gamma(G,pi) for the group of all automorphisms f of Fn for which pi*f = pi. This is called the standard congruence subgroup of Aut(Fn) associated to G and pi. In the case n = 2 we present formulas for the index of Gamma(G,pi) where G is abelian or dihedral. Moreover, we show that congruence subgroups associated to dihedral groups provide a family of subgroups of arbitrary large index in Aut(F2) generated by a fixed number of elements. This implies that finite index subgroups of Aut(F2) cannot be written as free products.
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