Geodesic systems of tunnels in hyperbolic 3–manifolds

Burton, Stephan D; Purcell, Jessica S

doi:10.2140/agt.2014.14.925

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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

DOI: 10.2140/agt.2014.14.925

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.

Keywords

tunnel systems, hyperbolic geometry, $3$–manifolds, geodesics

Mathematical Subject Classification 2010

Primary: 57M50

Secondary: 57M27, 30F40

References

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

Volume 14, issue 2 (2014)