Growth of Rees quotients of free inverse semigroups defined by small numbers of relators

D. Easdown and L.M. Shneerson

Abstract

We study the asymptotic behaviour of a finitely presented Rees quotient \(S={\rm Inv}\langle\, A\mid c_i=0 \;(i=1,\ldots, k)\,\rangle\) of a free inverse semigroup over a finite alphabet \(A\). It is shown that if the semigroup \(S\) has polynomial growth then \(S\) is monogenic (with zero) or \(k\ge 3\). The three relator case is fully characterised, yielding a sequence of two-generated three-relator semigroups whose Gelfand-Kirillov dimensions form an infinite set, namely \(\{4, 5, 6, \ldots\}\). The results are applied to give a best possible lower bound, in terms of the size of the generating set, on the number of relators required to guarantee polynomial growth of a finitely presented Rees quotient, assuming no generator is nilpotent. A natural operator is introduced, from the class of all finitely presented inverse semigroups to the class of finitely presented Rees quotients of free inverse semigroups, and applied to deduce information about inverse semigroup presentations with one or many relations. It follows quickly from Magnus' Freiheitssatz for one-relator groups that every inverse semigroup \(\Pi={\rm Inv}\langle \,a_1, \ldots, a_n\mid C=D\,\rangle\) has exponential growth if \(n>2\). It is shown that the growth of \(\Pi\) is also exponential if \(n=2\) and the Munn trees of both defining words \(C\) and \(D\) contain more than one edge.

Keywords: Rees quotient, free inverse semigroup, growth, Gelfand-Kirillov dimension.

AMS Subject Classification: Primary 20M18; secondary 20M05.