Asymptotic behavior of grafting rays

TL;DR

This paper investigates how grafting rays in Teichmüller space behave asymptotically, showing their convergence properties and relation to Teichmüller geodesics, especially along specific laminations and curve systems.

Contribution

It establishes the convergence behavior of grafting rays to the Thurston boundary and compares their distance to Teichmüller geodesics for certain laminations.

Findings

Grafting rays converge to the Thurston boundary similar to Teichmüller geodesics.

Grafted rays along weighted simple closed curves stay at bounded distance from Teichmüller geodesics.

Behavior matches that of lines of minima for specific laminations.

Abstract

In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmuller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmuller geodesics and lines of minima. We also show that the ray grafted along a weighted system of simple closed curves is at bounded distance from Teichmuller geodesic.