What are the applications of hyperbolic $3$-manifold theory to cosmology?

I am a pure mathematician specialized in hyperbolic $$3$$-manifold topology. That has been an incredibly active field of research in the past few decades due to the seminal work of Thurston, as many of his conjectures have been studied and resolved. I am curious about how this area can be used to inform astrophysics and especially cosmology, and what are the limitations.

One application I know about is the study of the shape of the general universe. Is that typically considered obsolete, since physicists seem to agree the universe is flat? Or are curved structures such as the Poincaré dodecahedral space still considered likely (for instance in the work of Jeff Weeks)?

Can hyperbolic $$3$$-manifolds, or sub-portions of them, also effectively model more local phenomenon? For instance, does it make sense to think of black holes like cusps of non-compact manifolds?

My colleagues usually use the upper half-space model or Poincaré ball model of hyperbolic space (sometimes the Klein model). As far as I know, in astrophysics and cosmology the Lobachevsky model is preferred, which only shows up rarely for us (there's the Epstein-Penner decomposition, but others are more obscure). Has this caused problems in transferring data between the two perspectives?

What other potential applications are we aware of for hyperbolic $$3$$-manifolds to astrophysics? (And if you think it is worth doing, what would be a good place to start reading, to make a transition?)

• The shape of the Universe is still a discussed topic but the general consensus is that the Universe is very close to flat (evidence provided by the 5-year WMAP data). – astronat Dec 6 '16 at 11:58
• Does it have to be cosmology? I think you'll find plenty of stuff in AdS/CFT. $AdS$ is just Lorentzian (signature 1 3) hyperbolic space, furthermore there are plenty of problems that concerns physics on a single "time slice", which has a Euclidean hyperbolic structure. here an example of a paper that points out (in an annoying physical way) how hyperbolic orbifolds show up in AdS/CFT. – zzz Dec 13 '16 at 20:49
• @astronat To me, that is like saying that, to a dust mite on a beach ball, the beach ball is "very close to flat." I've discussed this various times with various physicists and I can't seem to get a satisfying answer to this: how do we think we're measuring a large enough portion of the universe to detect its curvature? Every differentiable manifold is locally Euclidean and I'd bet the part we can see is pretty darn local. – j0equ1nn Mar 1 '19 at 4:03
• Well, we don't know we're measuring "enough". We can only measure what we can see. But the same would apply if we thought it was a Poincare dodecahedral space, and so I presume we go with spatially flat because Ockham's razor drives us to choose the simplest models where possible. – astronat Mar 2 '19 at 9:43
• @astronat For the sake of local navigation, etc, sure Euclidean is the simplest model (just like a local chart on an arbitrary manifold). But to extrapolate that to the entire universe sure seems foolish, and even is the exact same thinking behind the old theory that the Earth is flat. Moreover, I don't think an infinite universe is the simplest model, which globally Euclidean would imply. Nature tends to not like infinite. – j0equ1nn Mar 2 '19 at 21:34