Cosmetic surgery and the link volume of hyperbolic $3$–manifolds

Abstract

We prove that for any $V>0$ there exists a hyperbolic manifold $M_V$ such that $Vol\left(M_V\right)<2.03$ and $LinkVol\left(M_V\right)>V$. This was conjectured by the authors in [Algebr. Geom. Topol. 13 (2013) 927–958, Conjecture 1.3].

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 $ϵ>0$. Then all but finitely many hyperbolic manifolds obtained by filling $X$ admit a geodesic shorter than $ϵ$. (Note that it is not true that there are only finitely many fillings fulfilling this condition.)