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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
