Heegaard Gradient of Seifert Fibered 3-manifolds

The infimal Heegaard gradient of a 3-manifold was defined and studied by Marc Lackenby in an approach toward the well-known virtually Haken conjecture. As instructive examples, we consider Seifert fibered 3manifolds, and show that a compact orientable Seifert fibered 3-manifold has zero infimal Heegaard gradient if and only if it virtually fibers over the circle or over a surface other than the 2-sphere, equivalently, it has infinite fundamental group.