Group actions on complexes, Kozsul models, presentations, and a theorem of Coxeter

Abstract

By a well known theorem of K.S. Brown an action of a discrete group on a simply-connected complex allows to construct a presentation of this group modulo the stabilizers of vertices. The main goal of the present paper is to provide a new proof of this theorem based on ideas of J.-L. Kozsul. In contrast with Brown proof, we start with a redundant but fairly canonical presentation and then simplify it. We illustrate this procedure by the examples of fundamental groups of CW-complexes, the symmetric groups, the group of rotations of the regular dodecahedron, and the binary icosahedral group, and also discuss in details a remarkable theorem of Coxeter about the presentations of the latter group.