Infinite Reduced Words and the Tits Boundary of a Coxeter Group

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 ∂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 ξ, a natural order-preserving correspondence between infinite reduced words which “point towards” ξ, and elements of the reflection subgroup of W which fixes ξ.