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A Conjecture on the Hall Topology for the Free Group
Author(s) -
Pin JeanEric,
Reutenauer Christophe
Publication year - 1991
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/23.4.356
Subject(s) - mathematics , conjecture , free product , morphism , group (periodic table) , free group , discrete group , combinatorics , topology (electrical circuits) , simple group , finite group , closure (psychology) , alternating group , discrete mathematics , simple (philosophy) , pure mathematics , symmetric group , philosophy , chemistry , organic chemistry , epistemology , economics , market economy
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 H 1 , H 2 ,…, H n are finitely generated subgroups of the free group, then the product H 1 H 2 … H n 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.