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Oct 18, 2012 at 20:55 comment added HJRW Dave - not as far as I'm aware - indeed, we exhibit infinitely many commensurability classes of examples that are not commensurable with any 3-dimensional reflection group (see Section 6.3). So the above argument can't apply to those.
Oct 18, 2012 at 15:04 comment added Dave Futer Henry, thanks! So, in particular, are all the manifolds in your paper with Chesebro and DeBlois RFRS on the nose, without having to say "virtually"?
Oct 18, 2012 at 14:46 vote accept Dave Futer
Oct 18, 2012 at 14:41 comment added HJRW Minor edit: iirc, Agol also assumes that the preimages of the mirrors are non-separating in the manifold, so you may need to pass to an additional cover to ensure that this holds too.
Oct 18, 2012 at 14:32 history answered HJRW CC BY-SA 3.0