## Duke Mathematical Journal

### Large-scale rank of Teichmüller space

#### Abstract

Suppose that ${\mathcal{X}}$ is either the mapping class group equipped with the word metric or Teichmüller space equipped with either the Teichmüller metric or the Weil–Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that for any large box in ${\mathbb{R}}^{n}$ there is a standard model of a flat in ${\mathcal{X}}$ such that the quasi-Lipschitz image of a large sub-box is near the standard flat. As a consequence, we show that, for all these spaces, the geometric rank and the topological rank are equal. The methods are axiomatic and apply to a larger class of metric spaces.

#### Article information

Source
Duke Math. J., Volume 166, Number 8 (2017), 1517-1572.

Dates
Revised: 26 August 2016
First available in Project Euclid: 28 March 2017

https://projecteuclid.org/euclid.dmj/1490666574

Digital Object Identifier
doi:10.1215/00127094-0000006X

Mathematical Reviews number (MathSciNet)
MR3659941

Zentralblatt MATH identifier
1373.32012

#### Citation

Eskin, Alex; Masur, Howard; Rafi, Kasra. Large-scale rank of Teichmüller space. Duke Math. J. 166 (2017), no. 8, 1517--1572. doi:10.1215/00127094-0000006X. https://projecteuclid.org/euclid.dmj/1490666574

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