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# 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: http://dx.doi.org/10.4310/PAMQ.2015.v11.n3.a2

#### Authors

#### 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