Systoles and Dehn surgery for hyperbolic $3$–manifolds

Abstract

Given a closed hyperbolic $3$–manifold $M$ of volume $V$, and a link $L\subset M$ such that the complement $M\setminus L$ is hyperbolic, we establish a bound for the systole length of $M\setminus L$ 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 $7.35534\dots$ As part of the proof, we establish a bound for the systole length of a noncompact finite volume hyperbolic manifold which grows asymptotically like $\frac{4}{3}logV$.