text: abstract. we provide converses to two results of j. roe (geom. topol.
  2005): first, the warped cone over a free action of a haagerup group admits a
  fibred 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, e.g. for lp spaces.