Asymptoticity of grafting and Teichmüller rays

Subhojoy Gupta (Yale)

Abstract: We shall discuss a result showing that any grafting ray in Teichmüller space is (strongly) asymptotic to a Teichmüller geodesic ray. Our method involves constructing quasiconformal maps between the underlying Thurston metric of a complex projective surface on one hand, and the singular flat metric induced by a holomorphic quadratic differential on the other. One corollary is that the projection of a generic grafting ray to moduli space is dense.