A Conjecture on the Hall Topology for the Free Group
Jean-Éric Pin, Christophe Reutenauer · Bulletin of the London Mathematical Society · 1991
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.