Kazhdan-Lusztig cells in planar hyperbolic Coxeter groups and automata. (Q2923345)

Let \(W\) be a Coxeter group. If \(X\subseteq W\) define \(\mathrm{Red}(X)\) to be the set of all reduced expressions for elements of \(X\).NEWLINENEWLINE The main results that the authors obtain are the following:NEWLINENEWLINE (1) Let \(W\) be a word hyperbolic group, let \(S\) be any finite generating set for \(W\) satisfying \(S=S^{-1}\), and let \(\mu\) be any word in \(\mathrm{Red}(W)\). Then \(\mathrm{Red}(X_\mu)\) is a regular language.NEWLINENEWLINE (2) Let \(W\) be a word hyperbolic group, and let \(S\) be any generating set for \(W\). Suppose \(X\subseteq W\) is such that \(\mathrm{Red}(X)\) is a regular language. Then for any \(w\in W\), the language \(\mathrm{Red}(w\cdot X)\) is also regular.NEWLINENEWLINE The authors give two conjectures. Using these conjectures and the results above, they show that the language \(\mathrm{Red}(C)\) is regular for any left, right, or 2-sided Kazhdan-Lusztig cell \(C\).