Asian Journal of Mathematics

Volume 25 (2021)

Number 4

Metrics and compactifications of Teichmüller spaces of flat tori

Pages: 477 – 504



Mark Greenfield (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

Lizhen Ji (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R}) / \mathrm{SO}(n)$ with the Teichmüller space of flat $n$‑tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichmüller theory and symmetric spaces. We define and study analogs of the Thurston, Teichmüller, and Weil–Petersson metrics. We show the Teichmüller metric is a symmetrization of the Thurston metric, which is a polyhedral Finsler metric, and the Weil–Petersson metric is the Riemannian metric of $\mathrm{SL}(n,\mathbb{R}) / \mathrm{SO}(n)$ as a symmetric space. We also construct a Thurston-type compactification using measured foliations on $n$‑tori, and show that the horofunction compactification with respect to the Thurston metric is isomorphic to it, as well as to a minimal Satake compactification.


Teichmüller space, symmetric space, Riemann surface, flat torus

2010 Mathematics Subject Classification

30F60, 53C35

Received 20 June 2020

Accepted 26 November 2020

Published 25 April 2022