## 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 $$\xi$$.

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

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 Tuesday, January 8, 2013