Metrics and compactifications of Teichmüller spaces of flat tori

Abstract

Using the identification of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ with the Teichm\"uller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichm\"uller theory and symmetric spaces. We define and study analogs of the Thurston, Teichm\"uller, and Weil-Petersson metrics. We show the Teichm\"uller 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.