Combinatorial and Geometric Group Theory

Part of the series Trends in Mathematics pp 55-91

Geodesic Rewriting Systems and Pregroups

  • Volker DiekertAffiliated withUniversität Stuttgart
  • , Andrew J. DuncanAffiliated withNewcastle University
  • , Alexei G. MyasnikovAffiliated withMcGill University

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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