## Algebraic & Geometric Topology

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

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 2 (2014), 925-952.

Dates
Revised: 2 August 2013
Accepted: 6 September 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715853

Digital Object Identifier
doi:10.2140/agt.2014.14.925

Mathematical Reviews number (MathSciNet)
MR3160607

Zentralblatt MATH identifier
1286.57014

#### Citation

Burton, Stephan D; Purcell, Jessica S. Geodesic systems of tunnels in hyperbolic $3$–manifolds. Algebr. Geom. Topol. 14 (2014), no. 2, 925--952. doi:10.2140/agt.2014.14.925. https://projecteuclid.org/euclid.agt/1513715853

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