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# The Geometry of Carrier Graphs in Hyperbolic 3-Manifolds

thesis

posted on 2013-06-28, 00:00 authored by William M. SilerA 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## Usage metrics

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No categories selected## Licence

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