Abstract:
Until now, taut foliations on non-fibered hyperbolic 3-manifolds have generally been constructed using branched surfaces, whether via sutured manifold hierarchies, spanning surfaces of knot exteriors, or Dunfield's one-vertex triangulations with foliar orientations. In this talk, I introduce a novel taut foliation construction that makes no recourse to branched surfaces. Instead, it starts with a transversely-foliated $\mathbb{R}$-bundle over a Heegaard surface, specified by a real line action from a left-invariant order (when existent) on the fundamental group of the 3-manifold. Relating taut foliations to fundamental group real line actions is a decades old question that interested Thurston, Gabai, and Calegari, and has been revived in recent years by the L-space conjecture. This construction works reliably for genus 2 Heegaard diagrams satisfying mild conditions, explaining certain numerical results of Dunfield.