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

TL;DR

This paper provides a group-theoretic proof of the congruence subgroup property for Aut(F_2), simplifying Asada's original anabelian geometry approach and offering a quantitative version of the property.

Contribution

It translates Asada's geometric proof into a purely group-theoretic framework, making the proof more accessible and enabling quantitative analysis.

Findings

Simplified proof of the congruence subgroup property for Aut(F_2)

Quantitative version of the property established

Enhanced understanding of automorphism groups of free groups

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 $Aut(F_2)$.