Lengths of edges in carrier graphs

TL;DR

This paper investigates properties of minimal length carrier graphs in hyperbolic 3-manifolds, establishing conditions under which short edges imply the presence of short circuits, and broadening the class of manifolds with known minimal graphs.

Contribution

It proves that short edges in minimal length carrier graphs imply the existence of short circuits and extends the class of manifolds known to have minimal length carrier graphs.

Findings

Short edges in minimal carrier graphs imply short circuits.

The threshold for 'short' depends only on the fundamental group's rank.

Expanded the class of manifolds with known minimal length carrier graphs.

Abstract

We show that if X is a minimal length carrier graph in a hyperbolic 3-manifold, M, then if X contains a sufficiently short edge, it must contain a short circuit, as well. The meaning of "short" depends only on the rank of the fundamental group of M. We also expand the class of manifolds which are known to have minimal length carrier graphs.