Anosov representations: Domains of discontinuity and applications

TL;DR

This paper extends the concept of Anosov representations to all word hyperbolic groups, constructs domains of discontinuity, and explores their applications in geometric structures, compact forms, and space compactifications.

Contribution

It generalizes Anosov representations to arbitrary hyperbolic groups and systematically studies their geometric properties and applications.

Findings

Constructed explicit open sets with proper discontinuous group actions.

Showed higher Teichmüller spaces parametrize geometric structures on compact manifolds.

Derived applications to compact Clifford-Klein forms and space compactifications.

Abstract

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmueller spaces, and for lattices in SO(1,n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation of $\Gamma$ into G we explicitly construct open subsets of compact G-spaces, on which $\Gamma$ acts properly discontinuously and with compact quotient. As a consequence we show that higher Teichmueller spaces parametrize locally homogeneous geometric structures on compact manifolds. We also obtain applications regarding (non-standard) compact Clifford-Klein forms and compactifications of locally symmetric spaces of infinite volume.