Topological generation results for free unitary and orthogonal groups

Abstract

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