Infinite Reduced Words and the Tits Boundary of a Coxeter Group
Thomas Lam, Anne Thomas · International Mathematics Research Notices · 2014
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 |$\xi$|.