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The Geometry of Carrier Graphs in Hyperbolic 3-Manifolds

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posted on 28.06.2013, 00:00 by William M. Siler
A carrier graph is a map from a finite graph to a hyperbolic 3-manifold M, which is surjective on the level of fundamental groups. We can pull back the metric on M to get a notion of length for the graph. We study the geometric properties of the carrier graphs with minimal possible length. We show that minimal length carrier graphs exist for a large class of 3-manifolds. We also show that those manifolds have only finitely many minimal length carrier graphs, from which we deduce a new proof that such manifolds have finite isometry groups. Finally, we give a theorem relating lengths of loops in a minimal length carrier graph to the lengths of its edges. From this we are able, for example, to get an explicit upper bound on the injectivity radius of M based on the lengths of edges in a minimal length carrier graph.

History

Advisor

Shalen, Peter B.

Department

Mathematics, Statistics, and Computer Science

Degree Grantor

University of Illinois at Chicago

Degree Level

Doctoral

Committee Member

Groves, Daniel Culler, Marc Dumas, David Farb, Benson

Submitted date

2013-05

Language

en

Issue date

28/06/2013

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