Groups, Geometry, and Dynamics


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Volume 5, Issue 2, 2011, pp. 327–353
DOI: 10.4171/GGD/130

Published online: 2011-03-06

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

Kai-Uwe Bux[1], Mikhail Ershov[2] and Andrei S. Rapinchuk[3]

(1) Universität Bielefeld, Germany
(2) University of Virginia, Charlottesville, USA
(3) University of Virginia, Charlottesville, USA

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($F_2$).

Keywords: Automorphism groups, free groups, congruence subgroup property

Bux Kai-Uwe, Ershov Mikhail, Rapinchuk Andrei: The congruence subgroup property for Aut $F_2$: A group-theoretic proof of Asada’s theorem. Groups Geom. Dyn. 5 (2011), 327-353. doi: 10.4171/GGD/130