Hyperbolic configurations of roots and Hecke algebras

This paper deals with infinite Coxeter groups. We use geometric techniques to prove two main results. One is of Lie-theoretic nature; it shows the existence of many hyperbolic configurations of three pairwise disjoint roots in a given Coxeter complex, provided it is not an Euclidean tiling. The other is both of algebraic and measure-theoretic nature since it deals with Hecke algebras; it shows that for automorphism groups of buildings, convolution algebras of bi-invariant functions are never commutative, provided the building is not Euclidean. Proofs are of geometric nature: the main idea is to exhibit and use enough trees of valency ⩾3 inside a non-affine Coxeter complex.