Infinite reduced words and the Tits boundary of a Coxeter group

TL;DR

This paper explores the relationship between infinite reduced words in Coxeter groups and the topology of the Tits boundary of their Davis complexes, revealing structural insights especially in hyperbolic cases.

Contribution

It establishes a connection between the limit weak order of infinite reduced words and the Tits boundary topology, including special cases like hyperbolic Coxeter groups.

Findings

Limit weak order is encoded by Tits boundary topology.

In hyperbolic cases, the order decomposes into finite Coxeter group orders.

A natural correspondence exists between boundary points and reflection subgroup elements.

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 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.