Deligne's central extension is certainly an interesting group to consider. I can say something about the question whether this group is hyperlinear. For hyperlinearity, you ask for approximation of the group law by unitary matrices instead of permutations. This is (as you of course know) weaker than being sofic, but being sofic implies that the group is hyperlinear. Hence, it is a neccesary condition and natural generalization.

In the paper *Examples of hyperlinear groups without factorization property*, Groups Geom. Dyn. 4 (2010), no. 1, 195–208 I obverved that a central extension of a group $G$ by an abelian group $A$ is hyperlinear if and only if the twisted group von Neumann algebra $L_{\phi \circ \alpha} G$ is embeddable, where
$$\alpha \colon G \times G \to A$$ is the 2-cocycle which classifies the central extension, and $\phi$ belongs to a dense set in the Pontrjagin dual of $A$. This is the same as asking for a approximation of the group laws of $G$ with unitaries on a finite-dimensional Hilbert space, twisting the multiplication with the cocycle $\phi \circ \alpha$. The twisted setup is natural anyway and I propose to call a $S^1$-valued 2-cocycle on a group $G$ *hyperlinear* if you can find such an approximation by unitaries. It seems natural to consider the possibility that there are $S^1$-valued 2-cocycles even on residually finite groups which are not hyperlinear. However, I do not have any examples.

On the other side, if $G$ is residually finite and you can at the same time approximate $\phi \circ \alpha$ for sufficiently many $\phi$'s by suitable almost 2-cocycles defined on the finite quotients, you are in business. This would show that the central extension is at least hyperlinear.

I do not know about *sofic* in place of *hyperlinear*, since in the combinatorial setup, it is not possible to disintegrate the central extension over the Pontrjagin dual of $A$.

which is a central extension of a residually finite (indeed, linear) group. $\endgroup$ – HJRW Dec 8 '10 at 18:15