PDF《space》

1 Precisely the data are : a graph G on the set {1,r}, and for each edge {i, j} of G a natural number mij ≥ 2. When {i, j} is not an edge and i = j there is no relation involving si, sj - by convention we set mij = ∞. When i = j we set mij = 1. The pair (W, S = {s1,sr}) is called a Coxeter system. Every Coxeter group has the Haagerup property (see [4]). Many Coxeter groups are word-hyperbolic, but very few of these can be discrete cocompact in Hp, since their visual boundary is almost never a sphere. To be more precise a Coxeter group (W, S) is always a discrete cocompact group of automorphism of its Davis complex Σ(W, S) (see [7] for the construction of Σ(W, S) as a combinatorial object). Moussong proved that Σ(W, S) always admits a CAT(0) metric and proved that when W does not contain obvious free abelian groups of rank ≥ 2, then Σ(W, S) has a CAT(?1) metric, and thus W is wo........