A Lyndon's identity theorem for one-relator monoids

TL;DR

This paper proves that all one-relator monoids have favorable finiteness properties, constructs explicit resolutions, and classifies monoids with low cohomological and geometric dimensions, extending Lyndon's Identity Theorem to monoids.

Contribution

It establishes that one-relator monoids are of type ${ m FP}_ty$, constructs explicit projective resolutions, and classifies monoids with low cohomological and geometric dimensions.

Findings

All one-relator monoids are of type ${ m FP}_ty$.

Classification of one-relator monoids with cohomological dimension ≤ 2.

Description of the relation module as a principal left ideal.

Abstract

For every one-relator monoid $M = \langle A \mid u=v \rangle$ with $u, v \in A^*$ 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 ${\rm FP}_\infty$, 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 ${\rm F}_\infty$, and classifying the one-relator moniods with geometric dimension at most $2$. These results give a natural monoid analogue of Lyndon's Identity Theorem for one-relator groups.