Short geodesics in hyperbolic 3-manifolds

TL;DR

This paper proves that in hyperbolic 3-manifolds, sufficiently short geodesics are isotopic into a given Heegaard surface, establishing a link between geodesic length and topological embedding.

Contribution

It introduces a computable constant depending on genus, ensuring short geodesics can be isotoped into strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds.

Findings

Existence of a computable epsilon(g) for each genus g

Short geodesics are isotopic into the Heegaard surface

Results apply to complete hyperbolic 3-manifolds

Abstract

For each $g \ge 2$, we prove existence of a computable constant $\epsilon(g) > 0$ 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 $\epsilon(g)$ in $M$, then $\gamma$ is isotopic into $S$.