Discrete isometry groups of symmetric spaces

Abstract

This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $\Gamma <G$ of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different characterizations of the subclass of Anosov subgroups, which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of various equivalent dynamical and geometric properties (such as asymptotically embedded, RCA, Morse, URU). 3. Discuss the topological dynamics of discrete subgroups $\Gamma$ on flag manifolds associated to $G$ and Finsler compactifications of associated symmetric spaces $X=G/K$. Find domains of proper discontinuity and use them to construct natural bordifications and compactifications of the locally symmetric spaces $X/\Gamma$.

arXiv comment77 pages

References in corpus (8)

• A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings
• Rigidity of invariant convex sets in symmetric spaces
• Morse actions of discrete groups on symmetric space
• Algorithmic detection and description of hyperbolic structures on closed 3-manifolds with solvable word problem
• D-manifolds, d-orbifolds and derived differential geometry: a detailed summary
• Representations of knot groups into ${\mathrm{SL}}_n(\mathbf{C})$ and twisted Alexander polynomials
• Tameness of Riemannian locally symmetric spaces arising from Anosov representations
• Polyhedral Horofunction Compactification as a Polyhedral Ball

Cited by in corpus (4)

• Dynamics on flag manifolds: domains of proper discontinuity and cocompactness
• On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
• Relativizing characterizations of Anosov subgroups, I
• Complete affine manifolds with Anosov holonomy groups