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?)