We count the number of conjugacy classes of maximal, genus g , surface subroups in hyperbolic 3-manifold groups. For any closed hyperbolic 3-manifold, we show that there is an upper bound on this number which grows factorially with g . We also give a class of closed hyperbolic 3-manifolds for which there is a lower bound of the same type.
It is shown that the "standard double bubble" is the unique leastperimeter way to enclose and separate two given areas in the surface of the round sphere.
Let M be a once-punctured torus bundle over S 1 with monodromy h. We show that, under certain hypotheses on h, "most" Dehn-fillings of M (in some cases all but finitely many) are virtually Z-representable. We apply our results to show that even surgeries on the figure eight knot are virtually Z-representable.
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