Free wreath products as fundamental graph C*-algebras

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 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.