On some universal construction of minimal topological generating sets for inverse limits of iterated wreath products of non-Abelian finite simple groups

Adam Woryna

doi:10.1007/s10801-015-0584-3

Abstract

Let $$(H_i)_{i\ge 1}$$(Hi)i?1 be an arbitrary sequence of non-Abelian finite simple transitive permutation groups. By using the combinatorial language of time-varying automata, we provide an explicit and naturally defined construction of a two-element set which generates a dense subgroup in the inverse limit $$\ldots \wr H_2\wr H_1$$??H2?H1 of iterated permutational wreath products of the groups $$H_i$$Hi. The corresponding automaton is equipped with three states, one of which is neutral and the semigroup generated by the other two states is free. We derive other algebraic and geometric properties of the group generated by this automaton. By using the notion of a Mealy automaton, we obtain the analogous construction for the infinite permutational wreath power of an arbitrary non-Abelian finite simple transitive permutation group $$H$$H on a set $$X$$X. We show that the wreath power $$\ldots \wr H\wr H$$??H?H contains a dense 2-generated not finitely presented amenable subgroup of exponential growth, which is generated by a 3-state Mealy automaton over the alphabet $$X$$X. The self-similar group generated by this automaton is self-replicating, contracting and regular weakly branch over the commutator subgroup.

Highlights

Introduction and main resultsLet X be an arbitrary changing alphabet, i.e., an infinite sequence (Xi )i≥1 = (X1, X2, . . .) of non-empty finite sets and let X ∗ be the rooted tree defined by X
The automorphism group Aut (X ∗) is an example of a topological group equipped with its natural profinite topology induced by a metric in which two elements f, g ∈ Aut (X ∗) are close if they act in the same way on the set X t for a large value of t
Of iterated permutational wreath products of the groups Hi as a closed subgroup in the group Aut (X ∗) which consists of all automorphisms g such that for every vertex w ∈ X ∗, the restriction of g to the subtree {wx : x ∈ X|w|+1} ⊆ X ∗ (the so-called label of g at the vertex w, denoted further by g(w)) coincides with some element from H|w|+1, where |w| denotes the length of a word w

Summary

Introduction and main results

H H of an arbitrary non-Abelian finite simple transitive permutation group H on a set X and we take an arbitrary hooked triple (α, β, x) ∈ H × H × X of the group H In this case, the above construction brings a Mealy automaton over the alphabet X with the set of states (2) and the following system of wreath recursions:. Given a sequence (Hi )i≥1 of permutation groups on the sets Xi , we consider the closed subgroup W∞ of Aut (X ∗) that consists of all automorphisms g ∈ Aut (X ∗) satisfying: g(w) ∈ H|w|+1 for all w ∈ X ∗; we identify W∞ with the inverse limit of iterated permutational wreath products of the groups Hi : W∞ = .

Nearly finitary automorphisms and nearly finitary groups

Time-varying automata and groups generated by automata

Disappearing group words of an automaton

Automata with the nucleus

The proof of Theorem 1