The pro-nilpotent group topology on a free group

TL;DR

This paper studies the pro-nilpotent topology on free groups, providing algorithms for closure computations, and establishing decidability results for related algebraic structures.

Contribution

It introduces algorithms for computing closures in the pro-nilpotent topology and proves decidability of certain pseudovarieties related to free groups.

Findings

Closure of subgroup products is computable in the pro-nilpotent topology.

The nil-closure of rational subsets is computable.

Decidability of specific pseudovarieties involving G_nil.

Abstract

In this paper, we work on the pro-nilpotent group topology of a free group. First we investigate the closure of the product of finitely many subgroups of a free group in the pro-nilpotent group topology. We present an algorithm for the calculation of the closure in the pro-nilpotent group topology of the product of finitely many finitely generated subgroups of a free group. Then we deduce that the nil-closure of a rational subset is computable. We also prove that the pseudovarieties V malcev G_nil, for every decidable pseudovariety of monoids V, and J * G_nil are decidable.