I'm wondering if anyone knows how to *construct* hyperbolic 3-manifolds whose fundamental group is RFRS. Clearly the recent work of Agol, Wise, etc. says that such manifolds are abundant, and in particular present in every commensurability class. But how do you construct examples?


The only examples of RFRS manifolds that I'm aware of are torus knot complements (thanks to Stefan Friedl for pointing this out), but these are of course non-hyperbolic. 



Examples of 3-manifolds with RFRS fundamental group
Agol's original paper on RFRS gives a nice short proof that the fundamental group of any manifold which also happens to be a finite-index subgroup of your favourite right-angled reflection group is RFRS. (Sketch proof: watch how loops bounce off the mirrors!)


So take your favourite right-angled reflection group $\Gamma$ in $\mathbb{H}^3$ and take the commutator subgroup $[\Gamma,\Gamma]$ (which is always torsion-free, as all the torsion injects into $H\_1$). This will give you an example.


