Stretching three-holed spheres and the Margulis invariant

TL;DR

This paper demonstrates that infinitesimal deformations of a three-holed sphere that lengthen boundary components also lengthen all closed geodesics, using Lorentzian geometry and the Margulis invariant.

Contribution

It applies Lorentzian geometric methods to establish a new property of infinitesimal hyperbolic deformations of three-holed spheres.

Findings

Infinitesimal lengthening of boundary components implies lengthening of all closed geodesics.

The Margulis invariant interprets the derivative of geodesic length functions.

Properness of affine deformation follows from the lengthening property.

Abstract

This paper applies the authors' forthcoming work, "Affine deformations of a three-holed sphere" in Lorentzian geometry to prove a result in hyperbolic geometry. Namely, an infinitesimal deformation of a hyperbolic structure of a three-holed sphere which infinitesimally lengthens the three boundary components infinitesimally lengthens every closed geodesic. The proof interprets the derivative of the geodesic length function as the Margulis invariant (signed marked Lorentzian length spectrum) of the corresponding affine deformation. The aforementioned results imply that the affine deformation is proper, and hence by Margulis's Opposite Sign Lemma, every closed geodesic infinitesimmaly lengthens.