Combinatorial descriptions of multi-vertex $2$-complexes

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.