Duke Mathematical Journal

Large-scale rank of Teichmüller space

Alex Eskin, Howard Masur, and Kasra Rafi

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Suppose that 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 Rn there is a standard model of a flat in 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

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

Received: 17 September 2013
Revised: 26 August 2016
First available in Project Euclid: 28 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

rank coarse differentiation efficient hyperbolicity


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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