Wreath products with the integers, proper actions and Hilbert space   compression
  We prove that the properties of acting metrically properly on some space with walls or some CAT(0) cube complex are closed by taking the wreath product with \Z. We also give a lower bound for the (equivariant) Hilbert space compression of H\wr\Z in terms of the (equivariant) Hilbert space compression of H. 
Yves Stalder and Alain Valette