A Lyndon’s identity theorem for one-relator monoids

• Robert D. Gray ^[1] ; Benjamin Steinberg ^[2]
  1. [1] University of East Anglia

     University of East Anglia

     Norwich District, Reino Unido

     
  2. [2] City College of New York

     City College of New York

     Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 3, 2022
• Idioma: inglés
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• Resumen

  • For every one-relator monoid M = A | u = v with u, v ∈ A∗ we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type FP∞, answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property F∞, and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups.