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# Pure and Applied Mathematics Quarterly

## Volume 10 (2014)

### Number 3

### Characterization of the asymptotic Teichmüller space of the open unit disk through shears

Pages: 513 – 546

DOI: http://dx.doi.org/10.4310/PAMQ.2014.v10.n3.a4

#### Authors

#### Abstract

We give a parametrization to the asymptotic Teichmüuller space $AT(\mathbb{D})$ of the open unit disk $\mathbb{D}$ through equivalence classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tessellation of $\mathbb{D}$. Then using the parametrization, we define a new metric on $AT(\mathbb{D})$. Two other related metrics are also introduced on $AT(\mathbb{D})$ by using cross-ratio distortions or quadrilateral dilatations under the boundary maps on degenerating sequences of quadruples or quadrilaterals. We show that the topologies induced by the three metrics are equivalent to the one induced by the Teichmüller metric on $AT(\mathbb{D})$. Before proving our main results, we revisit and rectify a mistake in the proof in [21] on the characterization of quasisymmetric homeomorphisms in terms of shear functions.

#### Keywords

Teichmüller space, asymptotic Teichmüller space, extremal maximal dilatation, maximal quadrilateral dilatation, shear

#### 2010 Mathematics Subject Classification

30C75, 30F60