3-manifolding admitting locally large distance 2 Heegaard splittings

TL;DR

This paper investigates the conditions under which 3-manifolds with certain Heegaard splittings are hyperbolic, introducing the concept of locally large geodesics and characterizing hyperbolic cases among them.

Contribution

It introduces the notion of locally large geodesics in the curve complex and characterizes hyperbolic 3-manifolds with distance two Heegaard splittings.

Findings

Locally large distance two Heegaard splittings imply the manifold is hyperbolic or almost hyperbolic.

A non hyperbolic example with such splittings is constructed.

A necessary and sufficient condition for hyperbolicity in this context is provided.

Abstract

From the view of Heegaard splitting, it is known that if a closed orientable 3-manifold admits a distance at least three Heegaard splitting, then it is hyperbolic. However, for a closed orientable 3-manifold admitting only distance at most two Heegaard splittings, there are examples shows that it could be reducible, Seifert, toroidal or hyperbolic. According to Thurston's Geometrization conjecture, the most important piece of eight geometries is hyperbolic. Thus to read out a hyperbolic 3-manifold from a distance two Heegaard splittings is critical in studying Heegaard splittings. Inspired by the construction of hyperbolic 3-manifolds with a distance two Heegaard splitting [Qiu, Zou and Guo, Pacific J. Math. 275 (2015), no. 1, 231-255], we introduce the definition of a locally large geodesic in curve complex and furthermore the locally large distance two Heegaard splitting. Then we prove that if a 3-manifold admits a locally large distance two Heegaard splitting, then it is a hyperbolic manifold or an amalgamation of a hyperbolic manifold and a seifert manifold along an incompressible torus, i.e., almost hyperbolic, while the example in Section 3 shows that there is a non hyperbolic 3-manifold in this case. After examining those non hyperbolic cases, we give a sufficient and necessary condition for a hyperbolic 3-manifold when it admits a locally large distance two Heegaard splitting.