Every Coxeter group acts amenably on a compact space (Q2701845)

An action of a discrete group \(G\) on a compact space \(X\) is topologically amenable if there is a sequence of continuous maps \(b^n\colon X\to P(G)\) to the space of probability measures on \(G\) with the weak*-topology such that for every \(g\in G\), \(\lim_{n\to\infty}\sup_{x\in X}\|gb_x^n-b_{gx}^n\|_1=0\). Here the measure \(b_x^n=b^n(x)\) is considered as a function \(b_x^n\colon G\to[0,1]\) and \(\|\;\|_1\) is the \(l_1\)-norm. A discrete countable group \(G\) is called Higson-Roe amenable if \(G\) admits a topologically amenable action on a compact space. The asymptotic dimension of a metric space \(X\) does not exceed \(n\), if for arbitrary large \(d>0\) there are \(n+1\) uniformly bounded \(d\)-disjoint families \({\mathcal F}_i\) of sets in \(X\) such that the union \(\bigcup{\mathcal F}_i\) forms a cover of \(X\). A family \(\mathcal F\) is \(d\)-disjoint provided \(\min\{\text{dist}(x,y)\mid x\in F_1,\;y\in F_2,\;F_1\neq F_2,\;F_1,F_2\in{\mathcal F}\}\geq d\). In this paper, the authors prove that every Coxeter group \(\Gamma\) is Higson-Roe amenable and has finite asymptotic dimension.