A Conjecture on the Hall Topology for the Free Group

TLDR

The Hall topology on a free group is the coarsest making all homomorphisms onto finite discrete groups continuous, and every finitely generated subgroup is closed in this topology. We conjecture that the product of finitely many finitely generated subgroups is closed, and we discuss its implications for related conjectures, including Rhodes’ conjecture on finite semigroups. If true, this conjecture yields a simple algorithm for computing closures of rational subsets and implies a corresponding conjecture for the free monoid, which is equivalent to a deep finite‑semigroup conjecture for which J.

Abstract

The Hall topology for the free group is the coarsest topology such that every group morphism from the free group onto a finite discrete group is continuous. It was shoen by M. Hall Jr that every finitely generated subgroup of the free group is closed for this topology. We conjecture that if H1, H2,…,Hn are finitely generated subgroups of the free group, then the product H1 H2… Hn is closed. We discuss some consequences of this conjecture. First, it would give a nice and simple algorithm to compute the closure of a given rational subset of the free group. Next, it implies a similar conjecture for the free monoid, which in turn is equivalent to a deep conjecture on finite semigroups for the solution of which J. Rhodes has offered $100. We hope that our new conjecture will shed some light on Rhodes' conjecture.