A Lyndon’s identity theorem for one-relator monoids

Robert D. Gray, Benjamin Steinberg

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Abstract

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.
Original languageEnglish
Article number59
JournalSelecta Mathematica
Volume28
Issue number3
DOIs
Publication statusPublished - 27 Apr 2022

Keywords

  • Classifying space
  • Cohomological dimension
  • Geometric dimension
  • Homological finiteness property
  • One-relator monoid

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