Quasi-isometry and finite presentations of left cancellative monoids

TL;DR

This paper investigates geometric invariants of finitely generated left cancellative monoids, establishing that finite presentability and solvable word problem are quasi-isometry invariants, and provides a geometric characterization of finite presentability.

Contribution

It introduces a geometric characterization of finite presentability for left cancellative monoids and demonstrates its invariance under quasi-isometry, unlike in general monoids.

Findings

Finite presentability is a quasi-isometry invariant for left cancellative monoids.

Solvable word problem is also a quasi-isometry invariant in this class.

The geometric characterization does not extend to all monoids.

Abstract

We show that being finitely presentable and being finitely presentable with solvable word problem are quasi-isometry invariants of finitely generated left cancellative monoids. Our main tool is an elementary, but useful, geometric characterisation of finite presentability for left cancellative monoids. We also give examples to show that this characterisation does not extend to monoids in general, and indeed that properties such as solvable word problem are not isometry invariants for general monoids.