A remark by Gromov on 4-manifolds

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

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I don't understand that paragraph. He says that every 3-manifold is a quotient $S/\Gamma$ of a symmetric space by a discrete group of isometries $\Gamma$ quoting Thurston, but this fact is not true for all 3-manifolds (you need to remove essential spheres and tori), and one should quote Perelman instead of Thurston anyway. Then he says that in dimension $n\geqslant 4$ it is not known whether every manifold is homeomorphic to such a quotient $S/\Gamma$, but that looks not reasonable to me. What am I missing? – Bruno Martelli Aug 5 at 22:17
@Bruno, check "On universal groups and three-manifolds" by Hilden--Lozano--Montesinos--Whitten – Anton Petrunin Aug 5 at 22:35
aha, he says "possibly with fixed points!", I presumed $\Gamma$ was acting freely, a stupid mistake. Thanks – Bruno Martelli Aug 5 at 22:43

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 4-sphere (homeomorphic but not diffeomorphic to $S^4$), and getting a manifold which is homeomorphic but not diffeomorphic to a hyperbolic 4-manifold.

However, if the smooth Poincare conjecture is true, there still could be an exotic hyperbolic 4-manifold. The most powerful method for detecting non-diffeomorphic but homeomorphic 4-manifolds is the Donaldson or Seiberg-Witten invariants (it's conjectured that these are equivalent). It is also conjectured that these invariants are trivial on hyperbolic 4-manifolds. 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 4-manifold. Freedman-Quinn'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.

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Ian, I think, Gromov expects that every close PL 4-manifold is the quotient of the hyperbolic space by a discrete group with elliptic elements. Now that we know that every fp group could be the fundamental group of such quotient, maybe it is not such an outrageous expectation. If this is true then you get quotients that are homeo but not diffeo to each other. – Misha Aug 6 at 5:42
@Misha: How do we know that? – Bruno Martelli Aug 6 at 14:05
@Misha: Ok, I see, I also missed the part about allowing elliptic elements. – Agol Aug 6 at 14:40
@Bruno: This is a theorem by Panov and Petrunin: arxiv.org/abs/1104.4814 – Misha Aug 24 at 9:03