TITLE: Finite subgroups of unitary groups
QUESTION [25 upvotes]: Let $n$ be an integer. Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite group, one has $[u^m,v^m] = 1_n$. (In fact, he showed that any finite subgroup of $GL(n)$ has a abelian normal subgroup of finite index, bounded independently of the subgroup.)
In particular, the set of pairs that generate a finite subgroup cannot be dense in $U(n) \times U(n)$ for $n \geq 2$. Indeed, a generic pair of unitaries in $U(2)$ generates a free group. I am asking whether a stable version could still be true. 

Question  Let $n$ be an integer, let $u,v$ be a pair of unitaries in $U(n)$ and let $\varepsilon>0$. Is there some integer $k \in {\mathbb N}$ and unitaries $u',v' \in U(nk)$, such that  $\|u \otimes 1_k-u'\| \leq \varepsilon$ and $\|v \otimes 1_k - v'\| \leq \varepsilon$ (with respect to the standard embedding $U(n) \subset U(nk)$, $u \mapsto u \otimes 1_k$), and
  the unitaries $u',v'$ generate a finite group?

The question does only make sense once one has fixed a norm on $M_n {\mathbb C}$ for all $n \in {\mathbb N}$. I would either take the normalized Hilbert-Schmidt norm, i.e. $\|x\| = \frac1n Tr(x^*x)^{\frac12}$ for $x \in M_n \mathbb C$, or the operator norm.
(More globally seen, the questions are equivalent to the questions whether for the unitary group of certain UHF-algebras or the hyperfinite $II_1$-factor, pairs of unitaries generating a finite subgroup are dense in the natural topologies.)

REPLY [4 votes]: In the meantime I could answer the question myself; at least if one considers the metric induced by the norm. This will be contained in a forthcoming joint preprint with Ken Dykema. We can show two theorems:

Theorem 1: For every $\varepsilon>0$, there exists $n \in \mathbb N$ and unitaries $a,t \in U(n)$ such that $\|a-1\|\geq 1$ and
  $$\|1-tat^{-1}ata^{-1}t^{-1}a^{-2}\| < \varepsilon.$$

and

Theorem 2: There exists $\varepsilon_0>0$ such that for all $n \in \mathbb N$, and any two unitaries $a,t \in U(n)$ which generate a finite group: 
  $$\|1-tat^{-1}ata^{-1}t^{-1}a^{-2}\| < \varepsilon \leq \varepsilon_0$$
  implies
  $$\|1-a\|< 1/4-\sqrt{1/16-\varepsilon}.$$

This together implies that the answer to my question above is negative. The details are not complicated but a bit too long for putting them in this answer. I will post a link to the preprint as soon as it is ready.