 
Summary: RESEARCH BLOG 6/29/04
RANK VS. HEEGAARD GENUS
The rank of a group is the minimal number of generators. If one has
a closed manifold M3
of Heegaard genus g, then the fundamental group
of either handlebody of the Heegaard decomposition of M maps onto
1M, and therefore rank1M g (for manifolds with boundary, this
inequality is a bit more complicated). Sometimes this inequality is not
sharp for minimal genus Heegaard splittings. This is known for certain
Seifert fibred spaces and graph manifolds of rank two and Heegaard
genus 3 [2, 6], and in fact the Seifert fibred spaces for which the rank
and Heegaard genus differ have been classified. More recently, Schul
tens and Weidmann have shown that the rank and minimal Heegaard
genus can differ by arbitrary amounts for closed irreducible graph 3
manifolds (one may easily obtain reducible examples of this type by
taking connect sums of the BoileauZieschang examples). But there
is no known example of a closed hyperbolic 3manifold with rank and
Heegaard genus differing. The simplest special case is for manifolds
of rank two. I conjecture that these must have Heegaard genus two
(this doesn't assume the manifold is closed, only hyperbolic). This has
