Two-sided homological properties of special and one-relator monoids

TL;DR

This paper investigates the homological properties of special and one-relator monoids, establishing bounds on their cohomological dimensions and proving they are of type bi-FP_infinity.

Contribution

It relates the homological finiteness and Hochschild cohomological dimensions of monoids to those of their groups of units, and extends Lyndon's Identity theorem to these monoids.

Findings

Monoids with group of units of type FP_n also enjoy bi-FP_n.

Hochschild cohomological dimension of monoids is bounded by 2 or the dimension of the group of units.

All one-relator monoids of the form ⟨A | r=1⟩ are of type bi-FP_infinity.

Abstract

A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi-$\mathrm{FP}_n$ if its group of units is of type $\mathrm{FP}_n$. We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of 2 and the cohomological dimension of its group of units. We apply these results to prove a Lyndon's Identity type theorem for the two-sided homology of one-relator monoids of the form $\langle A \mid r=1 \rangle$. In particular, we show that all such monoids are of type bi-$\mathrm{FP}_\infty$. Moreover, we show that if $r$ is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most $2$, while if $r$ is a proper power then it has infinite Hochschild cohomological dimension. For any non-special one-relator monoid $M$ with defining relation $u=v$ we show that if there is no nonempty word $w$ such that $u,v \in A^*w \cap w A^*$ then $M$ is of type bi-$\mathrm{FP}_\infty$ and has Hochschild cohomological dimension at most $2$.