Short geodesics in hyperbolic $3$–manifolds

Abstract

For each $g\ge 2$, we prove existence of a computable constant $ϵ\left(g\right)>0$ such that if $S$ is a strongly irreducible Heegaard surface of genus $g$ in a complete hyperbolic $3$–manifold $M$ and $\gamma$ is a simple geodesic of length less than $ϵ\left(g\right)$ in $M$, then $\gamma$ is isotopic into $S$.