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 .
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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 .