#### Volume 14, issue 2 (2014)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Geodesic systems of tunnels in hyperbolic $3$–manifolds

### Stephan D Burton and Jessica S Purcell

Algebraic & Geometric Topology 14 (2014) 925–952
##### 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.

##### Keywords
tunnel systems, hyperbolic geometry, $3$–manifolds, geodesics
##### Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 57M27, 30F40