Combinatorial descriptions of multi-vertex 2-complexes

TL;DR

This paper introduces a method for efficiently describing multi-vertex 2-complexes using a variation of group presentations, facilitating geometric analysis of complex structures like torus knot groups and Artin groups.

Contribution

It presents a novel approach to describe multi-vertex 2-complexes through a slight modification of traditional group presentations, enabling easier derivation of their properties.

Findings

Describes multi-vertex 2-complexes for torus knot groups.

Provides a geometric proof of a classic result of Appel and Schupp.

Simplifies the analysis of elementary properties of complex groups.

Abstract

Group presentations are implicit descriptions of 2-dimensional cell complexes with only one vertex. While such complexes are usually sufficient for topological investigations of groups, multi-vertex complexes are often preferable when the focus shifts to geometric considerations. In this article, I show how to quickly describe the most important multi-vertex 2-complexes using a slight variation of the traditional group presentation. As an illustration I describe multi-vertex 2-complexes for torus knot groups and one-relator Artin groups from which their elementary properties are easily derived. The latter are used to give an easy geometric proof of a classic result of Appel and Schupp.