Grafting rays fellow travel Teichmuller geodesics

TL;DR

This paper proves that grafting rays in Teichmuller space behave like quasi-geodesics and stay close to actual geodesics, with controlled dependence on initial conditions, extending to quasi-Fuchsian groups.

Contribution

It establishes that grafting rays are quasi-geodesics and introduces a Lipschitz dependence of grafting on the starting point, extending to quasi-Fuchsian groups.

Findings

Grafting rays are quasi-geodesics in Teichmuller space.

Grafting maps are Lipschitz continuous with a universal constant.

Extension of grafting to neighborhoods in quasi-Fuchsian space.

Abstract

Given a measured geodesic lamination on a hyperbolic surface, grafting the surface along multiples of the lamination defines a path in Teichmuller space, called the grafting ray. We show that every grafting ray, after reparametrization, is a Teichmuller quasi-geodesic and stays in a bounded neighborhood of a Teichmuller geodesic. As part of our approach, we show that grafting rays have controlled dependence on the starting point. That is, for any measured geodesic lamination Lambda, the map of Teichmuller space which is defined by grafting along Lambda is L-Lipschitz with respect to the Teichmuller metric, where L is a universal constant. This Lipschitz property follows from an extension of grafting to an open neighborhood of Teichmuller space in the space of quasi-Fuchsian groups.