Systoles and Dehn surgery for hyperbolic 3-manifolds

TL;DR

This paper establishes bounds on the shortest closed geodesic (systole) length in hyperbolic 3-manifolds and their link complements, relating it to volume and extending previous results.

Contribution

It provides new bounds for systole lengths in hyperbolic 3-manifolds and their link complements, generalizing earlier bounds for non-hyperbolic cases.

Findings

Bound for systole length of M ackslash L in terms of volume V

Asymptotic growth of systole length as (4/3)log(V) for non-compact manifolds

Extension of Adams and Reid's universal bound to hyperbolic link complements

Abstract

Given a closed hyperbolic 3-manifold M of volume V, and a link L in M such that the complement M \ L is hyperbolic, we establish a bound for the systole length of M \ L in terms of V. This extends a result of Adams and Reid, who showed that in the case that M is not hyperbolic, there is a universal bound of 7.35534... . As part of the proof, we establish a bound for the systole length of a non-compact finite volume hyperbolic manifold which grows asymptotically like (4/3)log(V).