A note on the uniqueness of minimal length carrier graphs

TL;DR

This paper investigates the uniqueness of minimal length carrier graphs in hyperbolic 3-manifolds, showing finiteness and non-homotopy of such graphs in certain classes, and providing a new proof of finiteness of the isometry group.

Contribution

It establishes finiteness and non-homotopy of minimal length carrier graphs in a broad class of hyperbolic 3-manifolds, including geometrically finite ones.

Findings

Minimal length carrier graphs are not unique.

Finitely many minimal length carrier graphs exist in certain hyperbolic 3-manifolds.

The isometry group of a geometrically finite 3-manifold is finite.

Abstract

We show that minimal length carrier graphs are not unique, but if M is in a large class of hyperbolic 3-manifolds, including the geometrically finite ones, then M has only finitely many minimal length carrier graphs and no two of them are homotopic. As a corollary, we obtain a new proof that the isometry group of a geometrically finite 3-manifold is finite.