The Geometry of Carrier Graphs in Hyperbolic 3-Manifolds

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.