The Link Volume of Hyperbolic 3-Manifolds

TL;DR

This paper demonstrates that hyperbolic 3-manifolds can have arbitrarily large link volume while maintaining bounded volume, using cosmetic surgery techniques to explore the relationship between manifold volume and link complexity.

Contribution

It introduces new bounds and constructions relating hyperbolic volume and link volume, expanding understanding of manifold fillings and cosmetic surgeries.

Findings

Existence of hyperbolic manifolds with arbitrarily large link volume and bounded volume.

Most slopes on a link component do not yield cosmetic surgeries unless forming a Hopf link.

Finitely many fillings produce hyperbolic manifolds with short geodesics.

Abstract

We prove that for any V>0, there exist a hyperbolic manifold M_V, so that Vol(M_V) < 2.03 and LinVol(M_V) > V. The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statements, see the second part of the introduction. Here are two examples of the results we obtain: 1) Let K be a component of a link L in S^3. Then "most" slopes on K cannot be completed to a cosmetic surgery on L, unless K becomes a component of a Hopf link. 2) Let X be a manifold and \epsilon>0. Then all but finitely many hyperbolic manifolds obtained by filling X admit a geodesic shorter than \epsilon\ (note that this finite set may correspond to an infinitely many fillings).