Asymptoticity of grafting and Teichm\"{u}ller rays I

TL;DR

This paper proves that grafting rays in Teichmüller space are asymptotic to Teichmüller geodesic rays and demonstrates the density of certain grafting points and complex projective structures in moduli space.

Contribution

It establishes the asymptotic relationship between grafting and Teichmüller rays and shows the density of grafted points and projective structures in moduli space.

Findings

Grafting rays are asymptotic to Teichmüller geodesic rays.

The projection of generic grafting rays to moduli space is dense.

Integer graftings produce a dense set of points in Teichmüller space.

Abstract

We show that any grafting ray in Teichm\"{u}ller space determined by an arational lamination or a multi-curve is (strongly) asymptotic to a Teichm\"{u}ller geodesic ray. As a consequence the projection of a generic grafting ray to moduli space is dense. We also show that the set of points in Teichm\"{u}ller space obtained by integer (2\pi-) graftings on any hyperbolic surface projects to a dense set, which implies that complex projective surfaces with any fixed Fuchsian holonomy are dense in moduli space.