Topological generation results for free unitary and orthogonal groups

We show that for every N ≥ 3 the free unitary group UN+ is topologically generated by its classical counterpart UN and the lower-rank UN-1+. This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all N3, also hold at N = 3. Specifically, all discrete quantum duals UN+ and ON+ are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, UN+ are topologically generated by UN and their maximal tori - N (dual to the free groups on N generators) and similarly, ON+ are topologically generated by ON and their tori - .