University of Illinois at Chicago
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The Geometry of Carrier Graphs in Hyperbolic 3-Manifolds

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posted on 2013-06-28, 00:00 authored 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

2013-06-28

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