Anosov representations: domains of discontinuity and applications

By Olivier Y Guichard, Anna Wienhard. 2011

356 citations recorded by the source corpus.

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

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b74853d2a1bc6e4ea60ad29fb71cc0b87baff8e9

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Citations in the corpus

Invariant Measures for Horospherical Actions and Anosov Groups (2020, result): “We also refer to the work of Kapovich, Leeb and Porti [19] for other equivalent characterizations of Anosov groups, as well as to excellent survey articles by Kassel and Wienhard ([20], [45]) on higher Teichmüller theory.”
Conformality for a robust class of non-conformal attractors (2019, background): “Such representations provide the appropriate generalization of the class of convex co-compact subgroups in the context of Lie groups of higher rank [27, 31, 32].”
Anosov representations and dominated splittings (2016, background): “oups into semi-simple Lie groups, that unies examples of varying nature. Since then, Anosov representations have become a main object of study, being subject of various deep results (see for example [GW], the surveys [BCS,Wie] and references therein). Recently, new characterizations of Anosov repres”
Eigenvalues and entropy of a Hitchin representation (2014, methodology): “Note that we use the opposite convention than Guichard-Wienhard [19], our Pθ is their Pθc .”
The geometry of maximal representations of surface groups into SO0(2,n) (2017, methodology): “[GW12] construct a ρ-invariant open domain Ω in a certain flag manifold G/P on which ρ(Γ) acts properly discontinuously and co-compactly.”
Dynamics on flag manifolds: domains of proper discontinuity and cocompactness (2013, methodology): “Observe also that our treatment is intrinsic, while in [11], a theory for Anosov subgroups of Lie groups of the type Aut.”
AN INVITATION TO HIGHER TEICHMÜLLER THEORY (2019, unknown): “François Labourie, Jeremy Kahn, and Shahar Mozes (2017).”
Anosov representations and proper actions (2015, methodology): “ Pα0 is the stabilizer of an isotropic line, and by α1 the simple root of Gsuch that Pα1 is the stabilizer of a maximal isotropic subspace. The geometric construction of domains of discontinuity from [GW12] can be applied in many of the cases of Table 3 to furthermore obtain compactiﬁ- cations of th”
An invitation to higher Teichm\"uller theory (2018, methodology): “[64] For every split real simple Lie group G there exists a generalized flag variety X and a compact manifold M such that TH(S,G) parametrizes a connected component of the deformation space of (G,X)structures on M .”
The pressure metric for Anosov representations (2013, background): “Labourie and Sambarino were partially supported by the European Research Council under the European Community’s seventh Framework Programme (FP7/2007-2013)/ERC grant agreement n◦ FP7-246918, as well as by the ANR program ETTT (ANR-09BLAN-0116-01).”
Anosov groups: local mixing, counting and equidistribution (2020, methodology): “[19] O. Guichard and A. Wienhard.”
Hyperconvex representations and exponential growth (2012, result): “(Guichard and Wienhard [11, Proposition 4.10], Labourie [12, Proposition 2.1]) The function that associates to a strictly convex representation of its equivariant maps is continuous and the Hölder exponent of the equivariant maps can be chosen locally constant.”
Convex cocompactness in pseudo-Riemannian hyperbolic spaces (2017, methodology): “[14]); they sharemany dynamical properties with classical convex cocompact subgroups of rank-one simple Lie groups (see in particular [30,34,35,38]).”
Finsler bordifications of symmetric and certain locally symmetric spaces (2015, methodology): “24This means P mod –Anosov in the terminology of [24].”