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 ) whose ranks are one less than their genera. These were generalized by Scultens and Weidman  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  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.
 Boileau, M., Zieschang, H., Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math. 76 (1984), no. 3, 455–468.
 Schultens, Jennifer, Weidman, Richard, On the geometric and the algebraic rank of graph manifolds. Pacific J. Math. 231 (2007), no. 2, 481–510.