Hyperbolic Heegaard splittings and Dehn twists

Abstract

We consider the family of Heegaard splittings of genus $g$ at least two which are defined via a gluing 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 $1.24/(n^{2}g)$ and at most $37.5/n^2$

10 pages. 1 figure. V2: Improvements based on referee comments. Multiple constants improved; one factor-of-two error fixed. Comments welcome!