Topological generation results for free unitary and orthogonal groups

Abstract

We show that for every $N\ge 3$ the free unitary group $U^+_N$ is topologically generated by its classical counterpart $U_N$ and the lower-rank $U^+_{N-1}$. This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all $N\ne 3$, also hold at $N=3$. Specifically, all discrete quantum duals $\widehat{U^+_N}$ and $\widehat{O^+_N}$ are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, $U^+_N$ are topologically generated by $U_N$ and their maximal tori $\widehat{\mathbb{Z}^{*N}}$ (dual to the free groups on $N$ generators) and similarly, $O^+_N$ are topologically generated by $O_N$ and their tori $\widehat{\mathbb{Z}_2^{*N}}$.