Importance: High ✭✭✭
Author(s): Johnson, Jesse
Subject: Topology
Recomm. for undergrads: no
Posted by: Jesse Johnson
on: July 7th, 2008
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.


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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