The Haagerup property is not invariant under quasi-isometry

Abstract

Using the work of Cornulier-Valette and Whyte (see Appendix A for the latter), we show that neither the Haagerup property nor weak amenability is invariant under quasi-isometry of finitely generated groups. Appendix B shows, using the same examples, that the same holds for vanishing of the equivariant $L^p$-compression.

13 pages; Appendix A by Kevin Whyte; Appendix B by Sylvain Arnt, Thibault Pillon and Alain Valette. To appear in Bulletin London Math. Soc