On the Index of Congruence Subgroups of Aut(F_n)

TL;DR

This paper investigates the index of certain congruence subgroups of automorphism groups of free groups, providing formulas for specific cases and demonstrating the complexity of subgroup structures in Aut(F_2).

Contribution

It introduces formulas for the index of congruence subgroups in Aut(F_2) for abelian and dihedral groups and shows these subgroups can have arbitrarily large index.

Findings

Formulas for the index of Gamma(G,pi) when G is abelian or dihedral.

Construction of large index subgroups generated by few elements.

Finite index subgroups of Aut(F_2) are not free products.

Abstract

For an epimorphism pi of the free group F_n onto a finite group G write Gamma(G,pi) for the group of all automorphisms f of F_n for which pi*f = pi. This is called the standard congruence subgroup of Aut(F_n) 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(F_2) generated by a fixed number of elements. This implies that finite index subgroups of Aut(F_2) cannot be written as free products.