  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.
