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Rank vs. Genus
Question Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?
The rank of a 3-manifold is the minimal number of generators needed for its fundamental group. The Heegaard genus is the smallest genus of all Heegaard splittings for that 3-manifold. A Heegaard splitting determines a generating set for the 3-manifold, so the ranks is always less than or equal to the genus.
There is a family of Seifert fibered spaces for which the rank is one less than the genus, but for most Seifert fibered spaces, the rank and genus are equal. The Seifert fibered exampels have been used to construct graph manifolds for which the rank and genus differ by more than one [1]. However, there are no hyperbolic 3-manifolds for which rank and genus are known to differ.
Bibliography
Schultens, Jennifer, Weidman, Richard, On the geometric and the algebraic rank of graph manifolds. Pacific J. Math. 231 (2007), no. 2, 481--510.
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Connection to dynamics
Abert and Nikolov have found a connection between the `Rank vs Heegard Genus' problem and the `Fixed Price' problem in dynamics. Specifically, if every countable group has `fixed price' then the ratio of the Heegard genus and the rank of a hyperbolic 3-manifold can be arbitrarily large. For details, see http://arxiv.org/abs/math/0701361 .