Kazhdan-Lusztig cells in planar hyperbolic Coxeter groups and automata

TL;DR

This paper explores the structure of Kazhdan-Lusztig cells in hyperbolic Coxeter groups and conjectures their reduced expressions form regular languages, linking geometric group theory with automata theory.

Contribution

It provides a conjectural description of Kazhdan-Lusztig cells in hyperbolic polygon groups and connects these descriptions to the regularity of associated languages.

Findings

Conjectural descriptions of cells in hyperbolic Coxeter groups.

Implication that these descriptions support Casselman's conjecture.

Potential automata-theoretic characterization of cell-related languages.

Abstract

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.