Volume 14, issue 2 (2014)

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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
Publication
Received: 12 March 2013
Revised: 2 August 2013
Accepted: 6 September 2013
Published: 31 January 2014
Authors
 Stephan D Burton Department of Mathematics Michigan State University East Lansing, MI 48824 USA Jessica S Purcell Department of Mathematics Brigham Young University Provo, UT 84602-6539 USA