Ordering curves on surfaces

TL;DR

This paper investigates how the order of lengths of closed geodesics on hyperbolic surfaces relates to the geometry of the surface, showing that length orderings determine points in Teichmüller space and identifying invariant classes of curves.

Contribution

It establishes that length orderings of curves uniquely determine hyperbolic surface points and identifies classes of curves with invariant length orderings across metrics.

Findings

Length orderings determine Teichmüller space points.

Certain curve classes have invariant length orderings.

Short curves with small intersections are characterized.

Abstract

We study the order of lengths of closed geodesics on hyperbolic surfaces. Our first main result is that the order of lengths of curves determine a point in Teichm\"uller space. In an opposite direction, we identify classes of curves whose order never changes, independently of the choice of hyperbolic metric. We use this result to identify short curves with small intersections on pairs of pants.