Conformal limits of grafting and Teichm\"{u}ller rays and their asymptoticity

TL;DR

This paper proves that grafting rays in Teichmüller space are asymptotic to Teichmüller geodesic rays, introducing limit surfaces and using quasiconformal maps to establish the relationship.

Contribution

It establishes a general asymptotic relationship between grafting and Teichmüller rays, extending previous results to a broader class of rays with a unified approach.

Findings

Grafting rays are asymptotic to Teichmüller geodesic rays.

Introduction of limit surfaces from degenerating Riemann surfaces.

Construction of low dilatation quasiconformal maps between surfaces.

Abstract

We show that any grafting ray in Teichm\"{u}ller space is (strongly) asymptotic to some Teichm\"{u}ller geodesic ray. As an intermediate step we introduce surfaces that arise as limits of these degenerating Riemann surfaces. Given a grafting ray, the proof involves a Teichm\"{u}ller ray with a conformally equivalent limit, and building quasiconformal maps of low dilatation between the surfaces along the rays. Our preceding work had proved the result for rays determined by an arational lamination or a multicurve, and the unified approach here gives an alternative proof of the former case.