The congruence subgroup property for Aut $F_2$: A group-theoretic proof of Asada’s theorem

Abstract

The goal of this paper is to give a group-theoretic proof of the congruence subgroup property for Aut($F_2$), the group of automorphisms of a free group on two generators. This result was first proved by Asada using techniques from anabelian geometry, and our proof is, to a large extent, a translation of Asada’s proof into group-theoretic language. This translation enables us to simplify many parts of Asada’s original argument and prove a quantitative version of the congruence subgroup property for Aut($F_2$).