A Lyndon’s identity theorem for one-relator monoids

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 mathrm{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 mathrm{F}_infty , 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.