Effective bilipschitz bounds on drilling and filling

Abstract

We prove explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic $3$–manifold $N$ and the thick part of any of its long Dehn fillings. Given a bilipschitz constant $J>1$ and a thickness constant $𝜖>0$, we quantify how long a Dehn filling suffices to guarantee a $J$–bilipschitz map on $𝜖$–thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff’s theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infinitesimal change in metric during the cone deformation.

Our quantitative results have two immediate applications. First, we relate the Margulis number of $N$ to the Margulis numbers of its Dehn fillings. In particular, we give a lower bound on the systole of any closed $3$–manifold $M$ whose Margulis number is less than $0.29$. Combined with Shalen’s upper bound on the volume of such a manifold, this gives a procedure to compute the finite list of $3$–manifolds whose Margulis numbers are below $0.29$.

Our second application is to the cosmetic surgery conjecture. Given the systole of a one-cusped hyperbolic manifold $N$, we produce an explicit upper bound on the length of a slope involved in a cosmetic surgery on $N$. This reduces the cosmetic surgery conjecture on $N$ to an explicit finite search.