Dynamics of implicit operations and tameness of pseudovarieties of groups

Abstract

Abstract. This work gives a new approach to the construction of implicit operations. By considering "higher-dimensional" spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesskiȋ, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of p-groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups V, if it can be decided whether a finitely generated subgroup of the free group with the pro-V topology is dense, then V is tame.

Subword Complexity of Profinite Words and Subgroups of Free Profinite Semigroups

“…In particular, in view of the results of [3] we conclude that End Ω A V has a natural structure as a profinite monoid, something which is perhaps not entirely obvious although it could equally well be proved directly. Note also that, for w, π 1 , .…”

“…for all s j ∈ S . The set O(S m ), whose members are called (m-ary) implicit operators on S , is clearly a monoid under composition and it is observed in [3] that, in case S is finitely generated, it is in fact a profinite monoid with respect to the 1 -metric which is induced by the natural metric on components. In case S = Ω A S we will talk simply about implicit operators.…”

“…Each unary (monoid) implicit operation x → x ν has a natural interpretation on the monoid of implicit operators. Adopting a notational convention from [3], the i-th component of the implicit operator (π 1 , . .…”

“…The basic tool in this paper is the construction of profinite words via the infinite iteration of implicit operators, as introduced in [3]. Necessary and sufficient conditions are presented for this process, applied to the free generators of a free profinite semigroup, to produce free generators of a free profinite subgroup.…”

“…The paper consists of 7 sections. Section 1 recalls a few necessary facts about profinite semigroups and implicit operations, including some rudiments of the implicit operator approach from [3]. Section 2 begins with the study of one of the main objects on this paper, namely, that of a groupgeneric operator.…”