Characterizing a free group in its automorphism group

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 stan...

The Automorphism Tower of a Free Group

We prove that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof: the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F ). Introduction In 1975 J. Dyer and E. Formanek [2] had proved that the automorphism group of a finitely generated non-abelian free group F is complete (that i...

Square-free Rings and Their Automorphism Group

Finite-dimensional square-freeK-algebras have been completely characterized by Anderson and D’Ambrosia as certain semigroup algebras A ∼= KξS over a square-free semigroup S twisted by some ξ ∈ Z (S,K), a two-dimensional cocycle of S with coefficients in the group of units K∗ of K. D’Ambrosia extended the definition of square-free to artinian rings with unity and showed every square-free ring ha...

Characterizing the Multiplicative Group of a Real Closed Field in Terms of its Divisible Maximal Subgroup

Characterization of C by its Automorphism Group

Let M be a connected complex manifold of dimension n and let Aut(M) denote the group of holomorphic automorphisms of M . The group Aut(M) is a topological group equipped with the natural compact-open topology. We are interested in the problem of characterizing M by Aut(M). This problem becomes particularly intriguing when Aut(M) is infinite-dimensional. Let, for example, M = C and suppose that ...