We study stability properties of the Haagerup Property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking standard wreath products. Our construction also provides a characterization of subsets with relative Property T in a standard wreath product. . That special result appeared since then in the book [BrO]. We also refer to [CSV] for applications of the result in harmonic analysis. Here we mention another

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... pplication. It was asked in [Co1, 7.7(1)] whether the quotient of a Haagerup group by an amenable normal subgroup is always Haagerup; the answer is negative in a strong sense. Corollary 1.2. There exists a Haagerup group Γ 1 with a non-Haagerup quotient Proof. If Λ = SL 2 (Z) and ZΛ is its group ring, the standard wreath product Z SL 2 (Z) can be identified with ZΛ Λ, and Z 2 is a cyclic Λ-module, so is a quotient of the free cyclic Λ-module by some submodule N , and Z 2 Λ is the quotient of ZΛ Λ by the abelian normal subgroup N .