On the rank of a Coxeter group

TL;DR

This paper investigates the possible ranks of finitely generated Coxeter groups, establishing the conditions under which the rank varies and identifying all potential ranks for such groups.

Contribution

It characterizes all possible ranks of finitely generated Coxeter groups, clarifying when different generating sets lead to different ranks.

Findings

Finite rank iff the group is finitely generated

Infinite rank equals the size of the group

Finitely generated groups can have multiple ranks

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.