ascii X - Sun - Content - Lines : 405 EVERY COXETER GROUP ACTS AMENABLY ON A COMPACT SPACE

Coxeter groups are Higson-Roe amenable, i.e. they admit amenable actions on compact spaces. Moreover, they have finite asymptotic dimension. This answers affirmatively a question from [H-R]. 1. Higson-Roe amenability. An action of a discrete group G on a compact space X is topologically amenable [AD-R] if there is a sequence of continuous maps b n : X → P (G) to the space of probability measures on G with the weak *-topology such that for every g ∈ G, lim n→∞ sup x∈X gb n x − b n gx 1 = 0. Here a measure b n x = b n (x) is considered as a function b n x : G → [0, 1] and 1 is the l 1-norm. Definition. A discrete countable group G is called Higson-Roe amenable if G admits a topologically amenable action on a compact space or, equivalently, its natural action on the Stone-ˇ Cech compactification βG is topologically amenable. 2. Property A. The property A was introduced in [Yu]. For metric spaces of bounded geometry it was reformulated in [H-R] as follows. Property A. A discrete metric space Z has the property A if and only if there is a sequence of maps a n : Z → P (Z) such that (1) for every n there is some R > 0 with the property that for every z ∈ Z, supp(a n z) ⊂ {z ′ ∈ Z | d(z, z ′) < R} and (2) for every K > 0, lim n→∞ sup d(z,w)<K a n z − a n w 1 = 0. Lemma [H-R]. A finitely generated group G is Higson-Roe amenable if and only if the underlying metric space G with a word metric has property A. A tree T posesses a natural metric where every edge has the length one. We denote by V (T) the set of vertices of T with induced metric. The idea of the proof of the following proposition is taken from [Yu]. Proposition 1. For any tree T the metric space V (T) has property A. Proof. Let γ 0 : R → T be a geodesic ray in T , i.e. an isometric embedding of the half-line R. For every point z ∈ V (T) there is a unique geodesic ray γ z issued from z which intersects γ 0 along a geodesic ray. Let V = V (T) ∩ …