 
Summary: Illinois Journal of Mathematics
Volume 54, Number 1, Spring 2010, Pages 137154
S 00192082
COMBINATORIAL DESCRIPTIONS OF MULTIVERTEX
2COMPLEXES
JON MCCAMMOND
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, multivertex complexes are often preferable when the fo
cus shifts to geometric considerations. In this article, I show how
to quickly describe the most important multivertex 2complexes
using a slight variation of the traditional group presentation. As
an illustration, I describe multivertex 2complexes for torus knot
groups and onerelator 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.
Some cell complexes are easy to describe: a graph with one vertex corre
sponds to a set S indexing its edges and a onevertex combinatorial 2complex
can be constructed from an algebraic presentation S  R . When one tries
