Geodesic systems of tunnels in hyperbolic 3-manifolds

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 (1;n)–compression body with a system of n core tunnels, n−1 of which self-intersect.