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A conjecture on the Hall topology for the free group

By Jean-Eric Pin and Christophe Reutenauer

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 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

Topics: Profinite topology, Free group, finite semigroup, [MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR], [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
Publisher: London Mathematical Society
Year: 1991
OAI identifier: oai:HAL:hal-00019880v1
Provided by: Hal-Diderot
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