This is a fairly old problem that everyone has probably heard of, but it caused some controversy recently. Rank refers to the rank of the fundamental group of a 3-manifold, the minimal number of elements needed to generate the group. Genus is the Heegaard genus of the same 3-manifold, the minimum genus over all Heegaard splittings. For a closed 3-manifold the Heegaard splitting determines a number of generating sets for the fundamental group, and the number of generators is equal to the genus. Thus the rank of the fundamental group is less than or equal to the Heegaard genus. There is a single example of a Seifert fibered space, discovered by Boileau and Zieschang [1], whose rank is one less than its genus. Schultens later showed [2] how to glue together copies of this space to get a family of graph manifolds for which genus minus rank is arbitrarily large. For hyperbolic 3-manifolds, there are no known examples for which the rank and genus differ. Thus the question is: Is there a hyperbolic 3-manifold whose rank is strictly less than its genus? (And if so, how big can the difference be?)

The recent controversy stems from a preprint that appeared on the arXiv about a year ago. The preprint claimed to prove that there are hyperbolic 3-manifolds such that not only genus minus rank, but in fact genus divided by rank, is arbitrarily large. The proof used Lackenby’s work on property \tau (maybe I can talk about this in a future entry) to find a sequence of finite covers of a fixed 3-manifold such that the Heegaard genus grows linearly with the degree of the cover. They then showed (or tried to) that the rank for these 3-manifolds grows sub-linearly, implying that the rank and genus diverge by an unbounded amount. However, the method used to calculate the rank relied on a preprint (by different authors) which, it turned out, contained a number of problems and could not be verified. The authors of the rank vs. genus paper have since withdrawn it.

This asymptotic method is still valid as a potential approach to answering the rank vs. genus question, though it’s unclear if the property \tau techniques should be expected to bound genus better than they bound rank. For a non-assymptotic approach to the conjecture, Helen Wong (a recent student of Casson) has suggested a way [3] to use Reshetikhin-Turaev quantum invariants to differentiate rank and genus. These invariants can be calculated from a Dehn surgery diagram for the 3-manifold. Reshetikhin and Turaev showed that the resulting polynomial can be used to calculate a lower bound on the Heegaard genus of the resulting 3-manifold. Helen has shown that for the Boileau and Zieschang example, this bound is high enough to distinguish Heegaard genus from rank (3 vs. 2). Boileau and Zieschang’s original proof does not generalize to higher genus manifolds. (They show that the 3-manifold in question does not have an order two symmetry, so it can’t have a genus two Heegaard splitting.) The Reshetikhin-Turaev, on the other hand, could very well work for higher genus.

[1] Boileau, M. and Zieschang, H., Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math. 1987 76, 455-468.

[2] Schultens, Jennifer and Weidman, Richard On the geometric and the algebraic rank of graph manifolds. *Pacific J. Math.* 231 (2007), no. 2, 481–510.

[3] Follow link to arXiv.

There is a significant residue from the Abert–Nikolov paper you’re referring to. Despite the problems with the Dooley–Golodets paper they relied on, they have nevertheless shown that the Rank vs Heegard genus problem is connected to the Fixed Price problem in dynamics. The third version of their paper on the arXiv makes this clear.

Comment by Henry Wilton — July 13, 2008 @ 11:49 am |

Thanks for pointing that out. The updated version of their paper can be found here.

Comment by Jesse Johnson — July 14, 2008 @ 9:37 am |