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. In general, the definition makes sense for the free wreath product of a compact quantum group with a subgroup of a quantum automorphism group of a finite dimensional C*-algebra. In a joint work with Pierre Fima, we extended the definition to allow the second quantum group to be any compact quantum group acting on such a C*-algebra. We then studied what were the conditions on the action that ensured that approximation properties were preserved by taking free wreath product.
In this talk I will explain the construction of the usual free wreath product, and then our definition. I will then describe the conditions we need on the action in order to be able to show that the resulting quantum group has the approximation properties we are interested in. | 
