Subword Complexity of Profinite Words and Subgroups of Free Profinite Semigroups

Abstract

We study free profinite subgroups of free profinite semigroups of the same rank using, as main tools, iterated continuous endomorphisms, subword complexity, and the associated entropy. Main results include a general scheme to produce such subgroups and a proof that the complement of the minimal ideal in a free profinite semigroup on more than one generator is closed under all implicit operations that do not lie in the minimal ideal and even under their infinite iteration.

Profinite semigroups and applications

“…For the case of finitely generated relatively free profinite semigroups, the following result was already observed in [19] as a consequence of a result from [6]. We provide here a direct proof of the more general case without the assumption of relative freeness.…”

“…Yet very little is known about them in general, in particular for the finitely generated free profinite semigroups Ω A S. In this section we survey some recent results the author has obtained which reveal strong ties between Symbolic Dynamics and the structure of free profinite semigroups. See [9,8] for more detailed surveys and [19] for related work.…”

Complexity of the identity checking problem for finite semigroups

“…By the condition (c) and the equalities (1), the word w 2 1,k appears as a factor in the word w 1,k+1 . Hence in the semigroup S we have w 2 1,k ≥ J w 1,k+1 J w 1,k . Using Proposition 2.1.2, we conclude that the H -class H of the element w 1,k is a maximal subgroup of the semigroup S. Furthermore, in view of the condition (b) and the equalities (1), the word w 1,k appears as a prefix as well as a suffix of each of the words w i,k+1 , which, in turn, appear as factors in the word w 1,k+2 by (a).…”

“…Theorem 1 has arisen as one of the applications of the theory of group generic sets in the free profinite semigroup developed in [2]. In order to make the present paper be understandable without acquaintance with [2], we give here a "finitized" version of the proof in which all profinite objects are substituted by their suitable finite approximations.…”

“…In order to make the present paper be understandable without acquaintance with [2], we give here a "finitized" version of the proof in which all profinite objects are substituted by their suitable finite approximations. The reader who knows the definition and some basic properties of the free profinite semigroups can easily "pass to the limit" and recover the natural generality of the constructions presented below.…”