We provide the converses to two results of Roe [Warped cones and property A, Geom. Topol. 9 (2005) 163–178, doi:10.2140/9t.2005.9.163]: first, the warped cone associated to a free action of an a-T-menable group admits a fibered coarse embedding into a Hilbert space, and second, a free action yielding a warped cone with property A must be amenable. We construct examples showing that in both cases the freeness assumption is necessary. The first equivalence is obtained also for other classes of Banach spaces, in particular for Lp-spaces.

中文翻译：

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矫直扭曲的锥体

我们提供了 Roe [Warped cones and property A,几何。白杨。 9(2005) 163–178, doi:10.2140/9t.2005.9.163]：首先，与 aT-menable 群的自由作用相关的扭曲锥体允许纤维粗糙嵌入到希尔伯特空间中，其次，自由作用产生具有属性 A 的翘曲锥体必须是服从的。我们构建的例子表明，在这两种情况下，自由度假设都是必要的。对于其他类 Banach 空间，也可以获得第一个等价，特别是对于大号p-空格。