Finsler bordifications of symmetric and certain locally symmetric spaces

TL;DR

This paper links Finsler geometry with symmetric space compactifications, introducing bordifications for locally symmetric spaces, and characterizes Anosov subgroups via these geometric structures.

Contribution

It provides a geometric interpretation of Satake compactification using Finsler metrics and constructs bordifications for locally symmetric spaces, characterizing Anosov subgroups through these structures.

Findings

Maximal Satake compactification arises from Finsler horofunction boundary.

Constructs orbifold-with-corners bordifications for locally symmetric spaces.

Characterizes Anosov subgroups via existence of compactifications.

Abstract

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces $X/\Gamma$ for arbitrary discrete subgroups $\Gamma< G$. These bordifications result from attaching $\Gamma$-quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of Anosov subgroups. We show, conversely, that Anosov subgroups are characterized by the existence of such compactifications among uniformly regular subgroups. Along the way, we give a positive answer, in the torsion free case, to a question of Ha\"issinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.