Dynamics of implicit operations and tameness of pseudovarieties of groups

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 Zalesskiĭ, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of p p -groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups V \mathbf {V} , if it can be decided whether a finitely generated subgroup of the free group with the pro- V \mathbf {V} topology is dense, then V \mathbf {V} is tame.