Geodesic
Geodesic systems of tunnels in hyperbolic 3–manifolds
Burton, Stephan D1 ; Purcell, Jessica S2
1Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
2Department of Mathematics, Brigham Young University, Provo, UT 84602-6539, USA
Algebraic and Geometric Topology, Tome 14 (2014) no. 2, pp. 925-952
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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, 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.

DOI : 10.2140/agt.2014.14.925
Classification : 57M50, 57M27, 30F40
Keywords: tunnel systems, hyperbolic geometry, $3$–manifolds, geodesics

Burton, Stephan D  1   ; Purcell, Jessica S  2

1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
2 Department of Mathematics, Brigham Young University, Provo, UT 84602-6539, USA 
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Burton, Stephan D; Purcell, Jessica S. Geodesic systems of tunnels in hyperbolic 3–manifolds. Algebraic and Geometric Topology, Tome 14 (2014) no. 2, pp. 925-952. doi: 10.2140/agt.2014.14.925

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