Algebraic & Geometric Topology

Geodesic systems of tunnels in hyperbolic $3$–manifolds

Stephan D Burton and Jessica S Purcell

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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, n1 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, n1 of which self-intersect.

Article information

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

Received: 12 March 2013
Revised: 2 August 2013
Accepted: 6 September 2013
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57M27: Invariants of knots and 3-manifolds 30F40: Kleinian groups [See also 20H10]

tunnel systems hyperbolic geometry $3$–manifolds geodesics


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.

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