Topological finiteness properties of monoids, II: Special monoids, one-relator monoids, amalgamated free products, and HNN extensions

Abstract

We show how topological methods developed in a previous article can be applied to prove new results about topological and homological finiteness properties of monoids. A monoid presentation is called special if the right-hand side of each relation is equal to $1$. We prove results which relate the finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid inherits the finiteness properties ${\mathrm{F}}_n$ and ${\mathrm{FP}}_n$ from its group of units. We also obtain results which relate the geometric and cohomological dimensions of such a monoid to those of its group of units. We apply these results to prove a Lyndon’s Identity Theorem for one-relator monoids of the form $⟨A\mid r=1⟩$. In particular, we show that all such monoids are of type ${\mathrm{F}}_{\infty }$ (and ${\mathrm{FP}}_{\infty }$), and that when $r$ is not a proper power, then the monoid has geometric and cohomological dimension at most $2$. The first of these results, resolves an important case of a question of Kobayashi from 2000 on homological finiteness properties of one-relator monoids. We also show how our topological approach can be used to prove results about the closure properties of various homological and topological finiteness properties for amalgamated free products and HNN-extensions of monoids. To prove these results we introduce new methods for constructing equivariant classifying spaces for monoids, as well as developing a Bass–Serre theory for free constructions of monoids.