The link volume of $3$–manifolds

Abstract

We view closed orientable $3$–manifolds as covers of $S^3$ branched over hyperbolic links. To a cover $M\stackrel{p}{\to }{S}^3$, of degree $p$ and branched over a hyperbolic link $L\subset S^3$, we assign the complexity $pVol\left(S^3\setminus L\right)$. We define an invariant of $3$–manifolds, called the link volume and denoted by $LinkVol\left(M\right)$, that assigns to a 3-manifold $M$ the infimum of the complexities of all possible covers $M\to S^3$, where the only constraint is that the branch set is a hyperbolic link. Thus the link volume measures how efficiently $M$ can be represented as a cover of $S^3$.

We study the basic properties of the link volume and related invariants, in particular observing that for any hyperbolic manifold $M$, $Vol\left(M\right)$ is less than $LinkVol\left(M\right)$. We prove a structure theorem that is similar to (and uses) the celebrated theorem of Jørgensen and Thurston. This leads us to conjecture that, generically, the link volume of a hyperbolic $3$–manifold is much bigger than its volume.

Finally we prove that the link volumes of the manifolds obtained by Dehn filling a manifold with boundary tori are linearly bounded above in terms of the length of the continued fraction expansion of the filling curves.