Iterated Grafting and Holonomy Lifts of Teichmueller space

TL;DR

This paper investigates the behavior of iterated grafting in Teichmüller space, showing it approximates single multicurve grafting, and studies the boundedness and asymptotic limits of holonomy lifts and grafting sequences.

Contribution

It provides explicit descriptions of iterated grafting as a single multicurve and analyzes the boundedness and convergence properties of holonomy lifts and grafting sequences.

Findings

Iterated grafting is close to grafting along a single multicurve.

Holonomy lifts of Teichmüller geodesics have bounded Teichmüller distance.

Grafting sequences converge geometrically to a punctured surface.

Abstract

Let $X$ be a closed hyperbolic surface and $\lambda, \eta$ be weighted geodesic multicurves which are short on X. We show that the iterated grafting along $\lambda$ and $\eta$ is close in the Teichmueller metric to grafting along a single multicurve which can be given explicitly in terms of $\lambda$ and $\eta$. Using this result, we study the holonomy lifts $gr_{\lambda}\rho_{X,\lambda}$ of Teichmueller geodesics $\rho_{X,\lambda}$ for integral laminations $\lambda$ and show that all of them have bounded Teichmueller distance to the geodesic $\rho_{X,\lambda}$. We obtain analogous results for grafting rays. Finally we consider the asymptotic behaviour of iterated grafting sequences $\gr_{n\lambda}X$ and show that they converge geometrically to a punctured surface.