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Warped cones and property A. (English) Zbl 1082.53036
Let \(Y\) be a \(C^\infty\) compact manifold or a finite simplicial complex, \(\Gamma\) a finitely generated group acting on \(Y\). The warped cone \(O_\Gamma Y\) is the coarse space obtained by warping \(OY\) along the induced \(F\)-action. Let \(X\) be a bounded geometry proper metric space (its closed bounded subsets are compact) and \(\text{Prob}(X)\) denote the space of Radon probability measures on \(X\). \(X\) has property A if there exists a sequence of weak-\(*\) continuous maps \(f_n: X\to\text{Prob}(X)\) such that: 1) for each \(n\) there is an \(r\) such that, \(\forall x\), the measure \(f_n(x)\) is supported within \(B(x; r)\); 2) for each \(s> 0\), as \(n\to\infty\), \(\sup_{d(x,y)< s}\| f_n(x)- f_n(y)\|\to 0\).
The warped cone over on amenable action is a space with property A. Let \(\Gamma\) be a (discrete) group that acts on a compact Hausdorff space \(Y\). The action of \(\Gamma\) is amenable if there exists a sequence of weak-\(*\) continuous maps \(\mu_n: Y\to \text{Prob}(\Gamma)\) such that, \(\forall\gamma\in \Gamma\), \(\sup_{y\in Y}\|\gamma.\mu_n(y)- \mu_n(\gamma y)\|\to 0\) as \(n\to\infty\).
The author proves several results on spaces with and without property A. He used a constructive method. The main result is the following theorem: “Suppose that \(Y\) is a compact manifold, or a finite simplicial complex, and that the group \(\Gamma\) acts on \(Y\) amenably by Lipschitz homeomorphisms. Then the warped cone \(X= O_\Gamma(Y)\) has property A”. Some examples are also presented.

MSC:
53C20 Global Riemannian geometry, including pinching
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