Prising apart geodesics by length in hyperbolic 3-manifolds

TL;DR

This paper identifies conditions under which the length function of a curve in hyperbolic 3-manifolds and surfaces uniquely determines the curve, extending known results to more complex curves with self-intersections.

Contribution

It generalizes the length-determination property from simple curves to certain self-intersecting curves and boundary-homotopic curves in hyperbolic 3-manifolds.

Findings

Simple curves have length functions that determine them uniquely.

Curves with self-intersection number one also have this property, with one exception.

Curves homotopic to boundary simple curves in hyperbolizable 3-manifolds are determined by their length functions.

Abstract

In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely determines the curve. For an orientable surface $S$ of negative Euler characteristic, we extend the known result that simple curves have this property to curves with self-intersection number one (with one exceptional case on closed surfaces of genus two that we describe completely), while for hyperbolizable 3-manifolds, we show that curves freely homotopic to simple curves on $\partial M$ have this property.