10027/9909
William M. Siler
William M.
Siler
The Geometry of Carrier Graphs in Hyperbolic 3-Manifolds
University of Illinois at Chicago
2013
hyperbolic geometry
3-manifold
carrier graph
2013-06-28 00:00:00
Thesis
https://indigo.uic.edu/articles/thesis/The_Geometry_of_Carrier_Graphs_in_Hyperbolic_3-Manifolds/10955036
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.