A Heegaard splitting for a closed 3-manifold determines a presentation for the fundamental group of in which the number of generators is equal to the genus of the Heegaard splitting. This implies that the Heegaard genus of any 3-manifold is greater than or equal to its rank (i.e. the minimal number of elements in a generating set for its fundamental group.) The reverse inequality, however, does not hold: There are Seifert fibered spaces (discovered by Boileau and Zieschang [1]) whose ranks are one less than their genera. These were generalized by Scultens and Weidman [2] to graph manifolds in which the difference (genus – rank) is arbitrarily large, though the ratio (genus/rank) is bounded. Until recently, it was unknown if a rank/genus gap could occur in Hyperbolic manifolds. However, Tao Li has just posted a preprint [3] where he constructs hyperbolic examples in which the difference between rank and genus (though not the ratio) is arbitrarily large. This makes two existing questions much more intriguing: Is there a good characterization of hyperbolic manifolds with a rank/genus gap? Is there a bound on the ratio rank/genus?

Note: You may have seen a preprint a few years ago claiming to prove that rank/genus could be arbitrarily small. However, that paper turned out to rely on incorrect results from another paper, as I mentioned in an earlier post.

[1] Boileau, M., Zieschang, H., Heegaard genus of closed orientable Seifert 3-manifolds. *Invent. Math.* 76 (1984), no. 3, 455–468.

[2] 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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I asked Tao whether atoroidal books of I-bundles have rank=genus. He thought this should be true, but I’m wondering if anyone knows if this is proven somewhere?

Comment by Ian Agol — July 2, 2011 @ 7:27 pm |

Joseph Masters tried to prove rank=genus for rank two surface bundles with boundary. However, there is an error in his preprint on the arXiv that he hasn’t been able to fix. (Though he’s still working on it.) It may be known for some specific examples of books of I-bundles, but in general I think it’s unknown,

Comment by Jesse Johnson — July 2, 2011 @ 10:09 pm |

just a comment on the ratio mentioned in the above post. The genus/rank ration is bounded for graph manifolds and can also be bounded for 3-manifolds in general provided that there is such a bound for hyperbolic 3-manifolds.

Comment by Richard Weidmann — October 27, 2011 @ 3:31 am |

Thanks for pointing that out. Is this because there is a bound on how much the rank can drop under torus gluing? (And if so, is this written down somewhere?)

Comment by Jesse Johnson — October 28, 2011 @ 10:41 am |

There is a rank formula for acylindrical splittings similar to Grushko’s theom for free products, thus if you have a bound on the rank of the fundamental group of the manifold you can bound both the number of vertices of the underlying graph and the rank of the vertex groups. This is published for 1-acylindrical amalgamated products but generalizes to arbitrary k-acylindrical splittings.

Comment by Richard Weidmann — December 12, 2011 @ 11:55 am