Geodesic Bounds of Hyperbolic Link Complements in Hyperbolic $3$-Manifold

TL;DR

This paper establishes upper bounds on the lengths of the shortest closed geodesics in hyperbolic link complements within compact hyperbolic 3-manifolds, relating these bounds to the manifold's volume.

Contribution

It introduces a method to bound geodesic lengths in hyperbolic link complements based on the volume of the ambient manifold, providing new geometric constraints.

Findings

Upper bounds on geodesic lengths in terms of volume

Relations between link complement geometry and manifold volume

New geometric inequalities for hyperbolic 3-manifolds

Abstract

Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we try to establish an upper bound of the length of $n^{th}$ shortest closed geodesic of $M\setminus L$ in terms of $V$.