A Lyndon’s identity theorem for one-relator monoids

Gray, Robert D. and Steinberg, Benjamin (2022) A Lyndon’s identity theorem for one-relator monoids. Selecta Mathematica, 28 (3). ISSN 1022-1824

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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.

Item Type: Article
Additional Information: Funding Information: This work was supported by the EPSRC grant EP/N033353/1 ‘Special inverse monoids: subgroups, structure, geometry, rewriting systems and the word problem’. The second named author was supported by a PSC-CUNY award and the Fulbright Comission. This work was supported by the London Mathematical Society Research in Pairs (Scheme 4) grant (Ref: 41844), which funded a 6-day research visit of the second named author to the University of East Anglia (September 2019).
Uncontrolled Keywords: classifying space,cohomological dimension,geometric dimension,homological finiteness property,one-relator monoid,mathematics(all),physics and astronomy(all) ,/dk/atira/pure/subjectarea/asjc/2600
Faculty \ School: Faculty of Science > School of Mathematics
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Depositing User: LivePure Connector
Date Deposited: 07 Feb 2022 10:30
Last Modified: 27 May 2022 09:30
URI: https://ueaeprints.uea.ac.uk/id/eprint/83316
DOI: 10.1007/s00029-022-00773-3

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