  Let C be a one- or two-sided Kazhdan--Lusztig cell in a Coxeter group (W,S),
and let Reduced(C) denote the set of reduced expressions of all w in C,
regarded as a language over the alphabet S. Casselman has conjectured that
Reduced(C) is regular. In this paper we give a conjectural description of the
cells when W is the group corresponding to a hyperbolic polygon, and show that
our conjectures imply Casselman's.
