Communications in Analysis and Geometry
Volume 27 (2019)
$3$-manifolds admitting locally large distance $2$ Heegaard splittings
Pages: 1355 – 1379
It is known that every closed, orientable $3$-manifold admits a Heegaard splitting. By Thurston’s Geometrization conjecture, proved by Perelman, a $3$-manifold admitting a Heegaard splitting of distance at least $3$ is hyperbolic. So what about $3$-manifolds admitting distance at most $2$ Heegaard splittings?
Inspired by the construction of hyperbolic $3$-manifolds in [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 also a locally large distance $2$ Heegaard splitting. Then we prove that if a $3$-manifold admits a locally large distance $2$ Heegaard splitting, then it is either a hyperbolic $3$-manifold or an amalgamation of a hyperbolic $3$-manifold and a small Seifert fiber space along an incompressible torus. After examining those non hyperbolic cases, we give a sufficient and necessary condition to determine a hyperbolic $3$-manifold admitting a locally large distance $2$ Heegaard splitting.
Received 23 August 2016
Accepted 3 June 2017
Published 12 December 2019