Combinatorial descriptions of multi-vertex 2-complexes

Details Combinatorial descriptions of multi-vertex 2-complexes Combinatorial descriptions of multi-vertex 2-complexes

2009-09-25

arxiv.org/abs/0909.4774v1

Mathematics

Group Theory

18 pages, 7 figures

Scientific paper

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.

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