Journal of the Australian Mathematical Society

Table of Contents - 2008 - Volume , Issue 02  

Research Article

GROUPS AND SEMIGROUPS WITH A ONE-COUNTER WORD PROBLEM

DEREK F. HOLTa1 c1, MATTHEW D. OWENSa2 and RICHARD M. THOMASa3

a1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: D.F.Holt@warwick.ac.uk)

a2 Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

a3 Department of Computer Science, University of Leicester, Leicester LE1 7RH, UK (email: rmt@mcs.le.ac.uk)

Abstract

We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian.

(Received December 29 2007)

(Accepted May 21 2008)

2000 Mathematics subject classification

  • primary 20F10;
  • 20M45; secondary 68Q45;
  • 03D40

Keywords and phrases

  • groups;
  • semigroups;
  • word problem

Correspondence:

c1 For correspondence; e-mail: D.F.Holt@warwick.ac.uk

Footnotes

Dedicated to Cheryl Praeger for her sixtieth birthday