Congruence Subgroups of the Automorphism Group of a Free Group

Let n ≥ 2 and Fn be the free group of rank n. Its automorphism group Aut(Fn) has a well-known surjective linear representation ρ : Aut(Fn) −→ Aut(Fn/F ′ n) = GLn(Z) where F ′ n denotes the commutator subgroup of Fn. By Aut (Fn) := ρ(SLn(Z)) we denote the special automorphism group of Fn. For an epimorphism π : Fn → G of Fn onto a finite group G we call Γ(G, π) := {φ ∈ Aut(Fn) | πφ = π} the standard congruence subgroup of Aut(Fn) associated to G and π. These groups are the objects of our study, where we mainly focus on the case n = 2. Our most important results are the following. We fully describe the abelianization of Γ(G, π) ≤ Aut(F2) for abelian and dihedral groups G. We also show that standard congruence subgroups of Aut(F2) associated to dihedral groups provide a family of subgroups of Aut(F2) of increasing finite index while each is generated by four elements. This implies that finite index subgroups of Aut(F2) cannot be written as free products. Furthermore, we prove that standard congruence subgroups of Aut(F2) associated to finite non-perfect groups have infinite abelianization. We are also interested in the images of standard congruence subgroups of Aut(F2) under the representation ρ. For these we show that ρ(Γ(G, π)) ≤ SL2(Z) is a congruence subgroup, i.e., it contains a group of the form ker(SL2(Z) → SL2(Z/mZ)), whenever G is a finite metacyclic group. In the last chapter we discuss some open problems on standard congruence subgroups of Aut(F2) and give suggestions for further research.