Effective hyperbolization and length bounds for Heegaard splittings

Abstract

We consider 3-manifolds given as Heegaard splittings $M=H^{-}{\cup }_{\hat{I}}\pounds H^{+}$ with the aim to describe the hyperbolic metric of $M$ under topological conditions on the splitting guaranteeing that the manifold is hyperbolic. In particular, given a suitable "sufficiently incompressible" curve ${\hat{I}}^3\subset \hat{I}\pounds$, we show (without appealing to Geometrization) that $M$ is hyperbolic and we compute the length of ${\hat{I}}^3$ in terms of the projection coefficients of the disk sets, up to a uniform multiplicative error.

55 pages, 7 figures. Comments are welcome!