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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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References:
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