Compactification de Thurston d'espaces de r\'eseaux marqu\'es et de l'espace de Torelli

TL;DR

This paper introduces a new compactification of symmetric spaces of noncompact type, analogous to Thurston's compactification, and applies it to the Torelli space of hyperbolic surfaces, establishing isomorphisms with Satake compactifications.

Contribution

It defines a Thurston-type compactification for spaces of marked lattices and Torelli spaces, linking them to Satake compactifications and describing boundary stratifications.

Findings

The compactification is equivariantly isomorphic to a Satake compactification.

A new compactification of Torelli space is constructed and shown to be isomorphic to the Satake compactification.

The boundary stratification of the compactified space is explicitly described.

Abstract

We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichm\"uller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification of the Torelli space of a hyperbolic surface with marked points, and we show that it is equivariantly isomorphic to the Satake compactification of the image of the period mapping. Finally, we describe the natural stratification of a subset of the boundary.