## Abstract

We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R > 0 on injectivity radius, consider the set of points with injectivity radius at least R; we call this the R-thick part of the manifold. We show that for any > 0, there exists a knot K in the 3-sphere so that the ratio of the volume of the R-thick part of the knot complement to the volume of the knot complement is at least 1 − . As R approaches infinity, and as approaches 0, this gives a sequence of knots that is said to Benjamini–Schramm converge to hyperbolic space. This answers a question of Brock and Dunfield.

Original language | English |
---|---|

Pages (from-to) | 581-595 |

Number of pages | 15 |

Journal | Mathematical Research Letters |

Volume | 25 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jan 2018 |