On free wreath products of classical groups

We study the generalized free wreath product of classical groups introduced by the first author and Arthur Troupel. We give an explicit computation of the Haar state and deduce important properties of their associated operator algebra: in many cases, the von Neumann algebra is a full type ${\mathrm{I}\mathrm{I}}_1$-factor and the reduced C*-algebra is simple with unique trace.