Gromov remarks in a a survey on manifolds (p.12) that "it is hard to imagine that there are infinitely many nondiffeomorphic, but mutually homeomorphic, quotients of the hyperbolic 4space by discrete isometry groups". What is the background of that?

I'm not exactly sure what he has in mind. If the smooth Poincare conjecture is false in 4 dimensions, then one could imagine taking a connect sum with a fake 4sphere (homeomorphic but not diffeomorphic to $S^4$), and getting a manifold which is homeomorphic but not diffeomorphic to a hyperbolic 4manifold. However, if the smooth Poincare conjecture is true, there still could be an exotic hyperbolic 4manifold. The most powerful method for detecting nondiffeomorphic but homeomorphic 4manifolds is the Donaldson or SeibergWitten invariants (it's conjectured that these are equivalent). It is also conjectured that these invariants are trivial on hyperbolic 4manifolds. If this were true, then one would need a different technique for detecting the exotic smooth structure. Another technique for detecting exotic structures is the Rochlin invariant. There is a homotopy $S^3\times \mathbb{R}$ which contains an embedded smooth Poincare sphere, and therefore cannot be smoothly standard since the Poincare sphere cannot bound a contractible manifold since it has odd Rochlin invariant. So one could try to find a manifold homeomorphic to a hyperbolic manifold which contains a smoothly embedded Poincare sphere. There are two difficulties in finding such an example: creating the example, and proving that it is homeomorphic to a hyperbolic 4manifold. FreedmanQuinn's surgery theorem is not available for groups of exponential growth, so it's not clear what technique one would use for proving homemorphism, maybe some surgery which doesn't affect the homeomorphism type. 

