Geodesic systems of tunnels in hyperbolic $3$–manifolds

Abstract

It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite-volume hyperbolic $3$–manifold. In this paper, we address the generalization of this question to hyperbolic $3$–manifolds admitting tunnel systems. We show that there exist finite-volume hyperbolic $3$–manifolds with a single cusp, with a system of $n$ tunnels, $n-1$ of which come arbitrarily close to self-intersecting. This gives evidence that systems of unknotting tunnels may not be isotopic to geodesics in tunnel number $n$ manifolds. In order to show this result, we prove there is a geometrically finite hyperbolic structure on a $\left(1;n\right)$–compression body with a system of $n$ core tunnels, $n-1$ of which self-intersect.