Anosov representations: domains of discontinuity and applications

• 成果类型: Article
• 署名作者: Guichard, Olivier; Wienhard, Anna
• 署名单位: Princeton University; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0382-7
• 发表日期: 2012
• 页码: 357-438
• 关键词: surface group-representations; real projective-structures; clifford-klein forms; maximal representations; symmetric-spaces; teichmuller space; schottky-groups; cross ratios; convexes; MANIFOLDS

摘要

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 Teichmuller 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 I-> G we explicitly construct open subsets of compact G-spaces, on which I acts properly discontinuously and with compact quotient. As a consequence we show that higher Teichmuller 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.