Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 3

Topological characterization of an asymptotic Teichmüller space through measured geodesic laminations

Pages: 403 – 449

DOI: https://dx.doi.org/10.4310/PAMQ.2015.v11.n3.a2

Authors

Jinhua Fan (Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing China)

Jun Hu (Department of Mathematics, Brooklyn College of CUNY, Brooklyn, New York, U.S.A.; and Ph.D. Program in Mathematics, Graduate Center of CUNY, New York, N.Y., U.S.A.)

Abstract

Let $\mathbb{D}$ be the open unit disk in the complex plane $\mathbb{C}$ and centered at the origin, and let $\mathcal{ML}_{\mathcal{b}}(\mathbb{D})$ be the collection of Thurston bounded measured geodesic laminations on $\mathbb{D}$. We introduce an equivalence relation on $\mathcal{ML}_{\mathcal{b}}(\mathbb{D})$ such that the earthquake measure map induces a bijection between the asymptotic Teichmüller space $AT(\mathbb{D})$ and the quotient space $\mathcal{AML}_{\mathcal{b}}(\mathbb{D})$ of $\mathcal{ML}_{\mathcal{b}}(\mathbb{D})$ under the equivalence relation. Furthermore, we introduce a topology on $\mathcal{AML}_{\mathcal{b}}(\mathbb{D})$ under which the bijection is a homeomorphism between $AT(\mathbb{D})$ and $\mathcal{AML}_{\mathcal{b}}(\mathbb{D})$ with respect to the Teichmüller metric on $AT(\mathbb{D})$. Corresponding results are also developed for a bijection and then a homeomorphism between the tangent space $\mathcal{AZ}(\mathbb{S}^1)$ of $AT(\mathbb{D})$ at a base point and $\mathcal{AML}_{\mathcal{b}}(\mathbb{D})$ with respect to the asymptotic cross-ratio norm topology on $\mathcal{AZ}(\mathbb{S}^1)$ and the defined topology on $\mathcal{AML}_{\mathcal{b}}(\mathbb{D})$.

Keywords

earthquakes, Thurston bounded measured geodesic laminations, teichmüller spaces and asymptotic Teichmüller spaces

2010 Mathematics Subject Classification

30C75, 30F60

Published 29 November 2016