In the last three decades, both hyperbolic geometry and Floer homology have played a central role in the study of the geometry and topology of three-dimensional manifolds (see for example [1], [23], [38], [40], [65]). Despite this, and even though both subjects have by now reached their maturity, their mutual interaction (if any) remains extremely mysterious. For example, while the computation of the Floer homology for the Seifert fibered case is very well-understood in explicit, geometric terms [18], [55], the Floer homology of hyperbolic manifolds (i.e. admitting a metric with constant sectional curvature ' 1) has eluded similar descriptions. Because Mostow rigidity implies that the geometric invariants of a hyperbolic metric are indeed topological invariants, the following is a very natural yet outstanding problem one encounters.
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