Infinite Reduced Words and the Tits Boundary of a Coxeter Group

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 $\partial _T X$ 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 $\xi$.