Hyperbolic Heegaard splittings and Dehn twists

Abstract

We consider the family of Heegaard splittings of genus $g$ at least three which are defined via a glueing map that is the $n$-th power of the Dehn twist along a curve that satisfies a natural topological assumption, namely pared acylindricity. We show that if $n$ is at least 14, then the Heegaard splitting has a hyperbolic metric for which the simple closed curve defining the Dehn twist is a closed geodesic of length at least $0.7/(n^2g^2)$ and at most $34.3/n^2$.