The profinite topology of free groups and weakly generic tuples of automorphisms. Issue 4 (11th November 2021) - Sági, Gábor
Abstract: Let A be a countable first order structure and endow the universe of A with the discrete topology. Then the automorphism group Aut ( A ) of A becomes a topological group. A tuple of automorphisms ⟨ g 0, …, g n − 1 ⟩ is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the ⟨ g 0, …, g n − 1 ⟩ ‐orbit of each a ∈ A is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the point of view of finite combinatorics. The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if Aut ( A ) has finite topological rank r (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in Aut ( A ) r implies the existence of weakly generic tuples in Aut ( A ) n for all natural number n ≥ 1 . Finally, in Theorem 3.2 we show that if A is a countable model of an ℵ0 ‐categorical, simple theory in which all types over the empty set are stationary and A has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length. At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [11] about topological ranks of the automorphism groups of some structures.