[Paper Review] Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensions

Robert D. Gray, Benjamin Steinberg|arXiv (Cornell University)|May 9, 2018

Geometric and Algebraic Topology48 references4 citations

TL;DR

This paper establishes topological finiteness properties for monoids using equivariant classifying spaces and Bass–Serre theory, proving that special monoids inherit F_n and FP_n properties from their group of units. It resolves a long-standing question by showing one-relator monoids ⟨A∣r=1⟩ are of type F_∞ and have geometric and cohomological dimension at most 2 when r is not a proper power.

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 $F_n$ and $FP_n$ 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 $\langle A \mid r=1 angle$. In particular we show that all such monoids are of type $F_{\infty}$ (and $FP_{\infty}$), 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.

Motivation & Objective

• To develop a topological framework for studying finiteness properties of monoids, extending methods from group theory.
• To resolve Kobayashi's 2000 question on homological finiteness of one-relator monoids by proving they are of type F_∞.
• To establish closure properties for topological and homological finiteness under free constructions like amalgamated free products and HNN extensions of monoids.
• To generalize Bass–Serre theory to monoids and construct equivariant classifying spaces for monoid actions.
• To relate geometric and cohomological dimensions of special monoids to those of their group of units.

Proposed method

• Adapt topological methods from group theory, particularly Eilenberg–Mac Lane spaces and K(G,1) complexes, to monoids.
• Construct equivariant classifying spaces for monoids using discrete group actions and monoid actions on simplicial complexes.
• Introduce a monoid version of Bass–Serre theory using trees and forests to model free constructions.
• Use flatness and projectivity conditions on group and monoid rings to transfer finiteness properties across extensions.
• Apply the Anick–Groves–Squier Theorem and Hochschild cohomology to relate rewriting systems and FP_n properties.
• Establish criteria for monoids to be of type F_n or FP_n based on the structure of their group of units and submonoid actions.

Experimental results

Research questions

• RQ1 Do special monoids inherit the F_n and FP_n properties from their group of units?
• RQ2 Are one-relator monoids of the form ⟨A∣r=1⟩ of type F_∞, and what are their geometric and cohomological dimensions?
• RQ3 Can the theory of amalgamated free products and HNN extensions be extended to monoids with analogous finiteness properties?
• RQ4 What conditions ensure that a monoid extension preserves topological or homological finiteness properties?
• RQ5 How do geometric and cohomological dimensions of special monoids relate to those of their group of units?

Key findings

• Special monoids inherit the F_n and FP_n properties from their group of units, establishing a strong link between monoid and group finiteness properties.
• One-relator monoids of the form ⟨A∣r=1⟩ are of type F_∞ (and FP_∞), resolving a key open problem posed by Kobayashi in 2000.
• When the relator r is not a proper power, such monoids have geometric and cohomological dimension at most 2.
• Amalgamated free products of monoids preserve F_n and FP_n properties under conditions on the monoid and submonoid actions.
• HNN-like extensions of monoids preserve bi-FP_n and bi-F_n properties when the base monoid and submonoid satisfy flatness and finiteness conditions.
• The geometric dimension of an HNN extension is bounded by the maximum of the base monoid’s geometric dimension and one more than the submonoid’s, under freeness assumptions.