Spacious knots (Q1990270)

The paper is motivated by the question of which 3-manifolds can occur as geometric limits of hyperbolic knot complements in $S^3$. \textit{J. S. Purcell} and \textit{J. Souto} [J. Topol. 3, No. 4, 759--785 (2010; Zbl 1267.57022)] showed that any one-ended hyperbolic 3-manifold that embeds into $S^3$ is a geometric limit of hyperbolic knot complements, including hyperbolic 3-space $\mathbb H^3$ itself; on the other hand, \textit{R. P. Kent IV} and \textit{J. Souto} [Math. Z. 271, No. 1--2, 565--575 (2012; Zbl 1282.57020)] presented compact submanifolds of $S^3$ whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements. Since $\mathbb H^3$ is a geometric limit of knot complements, for any $R>0$ there exists a hyperbolic knot $K$ and a point in $S^3 - K$ with injectivity radius at least $R$. Recently \textit{J. F. Brock} and \textit{N. M. Dunfield} [Geom. Topol. 19, No. 1, 497--523 (2015; Zbl 1312.57022)] asked if, for any $R$, there exists a sequence of hyperbolic knots for which the proportion of the volume of the $R$-thin part (i.e., all points with injectivity radius smaller than $R$) tends to zero (this is called \textit{Benjamini-Schramm convergence to} $\mathbb H^3$); in the present paper, this is answered in the affirmative. In particular, performing $1/n$-surgeries on these knots produces new examples of homology 3-spheres Benjamini-Schramm converging to $\mathbb H^3$.