A conjecture on the Hall topology for the free group

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 shown 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 H1H2 · · · 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 semigroup, for the solution of which J. Rhodes has offered $100. We hope that our new conjecture will shed some light on the Rhodes’s conjecture. 1 A conjecture on the Hall topology for the free group Let A be a finite set, called the alphabet. We denote by A the free monoid over A, and by F (A) the free group over A. The identity of F (A) is denoted by 1. The Hall (or profinite) topology for the free group was introduced by M. Hall in [6]. It is the coarsest topology on F (A) such that every group morphism from F (A) onto a finite discrete group is continuous. That is, a sequence un of elements of F (A) converges to an element u of F (A), if and only if, for every group morphism φ : F (A) → G (where G is a finite group), there exists an integer nφ such that for every n ≥ nφ, φ(un) = φ(u). This topology can also be defined by a distance. Since the free group is residually finite, two distinct elements u and v of the free group can always be “separated” by a finite group. More precisely, there exists a group morphism LITP, Tour 55-65, Université Paris VI et CNRS, 4 Place Jussieu, 75252 Paris Cedex 05, FRANCE UQAM, Dept. Math. Informatique Case Postale 8888, succursale A, Montréal, Québec H3C 3P8 CANADA