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

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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.