Infinite reduced words and the Tits boundary of a Coxeter group

Thomas Lam and Anne Thomas

Abstract

Let \((W,S)\) be a finite rank Coxeter system with \(W\) infinite. We prove that the limit weak order on the blocks of infinite reduced words of \(W\) is encoded by the topology of the Tits boundary \(\partial_TX\) of the Davis complex \(X\) of \(W\). We consider many special cases, including \(W\) word hyperbolic, and \(X\) with isolated flats. We establish that when \(W\) is word hyperbolic, the limit weak order is the disjoint union of weak orders of finite Coxeter groups. We also establish, for each boundary point \(\xi\), a natural order-preserving correspondence between infinite reduced words which "point towards" \(\xi\), and elements of the reflection subgroup of \(W\) which fixes \(\xi\).

AMS Subject Classification: Primary 20F55; secondary (primary), 52C35, 20F65 (secondary).