The free wreath product of a compact quantum group by the quantum permutation group S+N has been introduced by Bichon in order to give a quantum counterpart of the classical wreath product. The representation theory of such groups is well-known, but some results about their operator algebras were still open, for example, the Haagerup property, K-amenability, or factoriality of the von Neumann algebra. I will present a joint work with Pierre Fima in which we identify these algebras with the fundamental 𝐶^∗-algebras of certain graphs of 𝐶^∗-algebras, and we deduce these properties from these constructions
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