Geodesic Rewriting Systems and Pregroups

  • Volker Diekert
  • Andrew J. Duncan
  • Alexei G. Myasnikov
Part of the Trends in Mathematics book series (TM)


In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at our main object of study which we call geodesically perfect rewriting systems. We show that these are well behaved and convenient to use, and give several examples of classes of groups for which they can be constructed from natural presentations. We describe a Knuth-Bendix completion process to construct such systems, show how they may be found with the help of Stallings’ pregroups and conversely may be used to construct such pregroups.

Mathematics Subject Classification (2000)

68Q42 20F05 20M32 20E06 


String rewriting systems Geodesically Perfect Knuth-Bendix Stallings pregroups 


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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Volker Diekert
    • 1
  • Andrew J. Duncan
    • 2
  • Alexei G. Myasnikov
    • 3
  1. 1.Universität StuttgartStuttgartGermany
  2. 2.Newcastle UniversityNewcastle upon TyneUK
  3. 3.McGill UniversityMontrealCanada

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