Complete positive group presentations

TL;DR

This paper introduces the concept of completeness in positive group presentations, providing criteria and methods to recognize and complete such presentations, which helps analyze properties of the associated monoids and groups.

Contribution

It defines the property of completeness for positive group presentations and offers an effective criterion and iterative method for recognizing and completing them.

Findings

Complete presentations allow reading properties like cancellativity and common multiples.

A new criterion for monoid embedding into groups of fractions is established.

Applicable to standard presentations of Artin groups and the Heisenberg group.

Abstract

A combinatorial property of prositive group presentations, called completeness, is introduced, with an effective criterion for recognizing complete presentations, and an iterative method for completing an incomplete presentation. We show how to directly read several properties of the associated monoid and group from a complete presentation: cancellativity or existence of common multiples in the case of the monoid, or isoperimetric inequality in the case of the group. In particular, we obtain a new criterion for recognizing that a monoid embeds in a group of fractions. Typical presentations eligible for the current approach are the standard presentations of the Artin groups and the Heisenberg group.