Topological generation and matrix models for quantum reflection groups

We prove that the von Neumann algebras of quantum permutation groups and quantum reflection groups have the Connes embedding property. We do this by establishing several new topological generation results for the quantum permutation groups S_N ⁺ and the quantum reflection groups H_N ^s+. We use these results to show that these quantum groups admit sufficiently many “matrix models”. In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain “flat” matrix models for S_N ⁺ are inner faithful.