Straightening warped cones

Abstract

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 [Formula: see text]-spaces.