On the Abelianization of Congruence Subgroups of Aut(F_2)

TL;DR

This paper investigates the abelianization of certain automorphism subgroups of free groups, providing a complete description for finite abelian groups and revealing infinite abelianization for non-perfect groups.

Contribution

It fully characterizes the abelianization of congruence subgroups of Aut^+(F_2) for finite abelian G and shows infinite abelianization for non-perfect G.

Findings

Complete description of abelianization for finite abelian G

Infinite abelianization for non-perfect G

Advances understanding of automorphism subgroup structures

Abstract

Let F_n be the free group of rank n and let Aut^+(F_n) be its special automorphism group. For an epimorphism pi : F_n -> G of the free group F_n onto a finite group G we call Gamma^+(G,pi) = {f in Aut^+(F_n) | pi*f = pi} the standard congruence subgroup of Aut^+(F_n) 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^+(F_2) has infinite abelianization.