## The Geometry of Carrier Graphs in Hyperbolic 3-Manifolds

thesis

posted on 28.06.2013 by William M. Siler#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

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.