On the rank of a Coxeter group
Group Theory
2007-06-28 v1
Abstract
Let W be a Coxeter group with Coxeter generators S. The rank of the Coxeter system (W,S) is the cardinality |S| of S. The Coxeter system (W,S) has finite rank if and only if W is finitely generated. If (W,S) has infinite rank, then |S| = |W|, since every element of W is represented by a finite product of elements of S. Thus if W is not finitely generated, the rank of (W,S) is uniquely determined by W. If W is finitely generated, then W may have sets of Coxeter generators S and S' of different ranks. In this paper, we determine the set of all possible ranks for an arbitrary finitely generated Coxeter group W.
Keywords
Cite
@article{arxiv.0706.3911,
title = {On the rank of a Coxeter group},
author = {Michael L. Mihalik and John G. Ratcliffe},
journal= {arXiv preprint arXiv:0706.3911},
year = {2007}
}