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Illinois Journal of Mathematics Volume 54, Number 1, Spring 2010, Pages 137154
 

Summary: Illinois Journal of Mathematics
Volume 54, Number 1, Spring 2010, Pages 137154
S 0019-2082
COMBINATORIAL DESCRIPTIONS OF MULTI-VERTEX
2-COMPLEXES
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, multi-vertex complexes are often preferable when the fo-
cus 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.
Some cell complexes are easy to describe: a graph with one vertex corre-
sponds to a set S indexing its edges and a one-vertex combinatorial 2-complex
can be constructed from an algebraic presentation S | R . When one tries

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics