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Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces

Abstract (Expand)

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the … target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite dimensional totally geodesic subspace on which the action is maximal. In the opposite direction we construct examples of geometrically dense maximal representation in the infinite dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, that we are able to construct in low ranks or under some suitable Zariski-density assumption, circumventing the lack of local compactness in the infinite dimensional setting.

Authors: Bruno Duchesne, Jean Lécureux, Maria Beatrice Pozzetti

Date Published: 2018

Publication Type: Journal

Hitchin components for orbifolds

Abstract (Expand)

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold … group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on 3-manifolds.

Authors: Daniele Alessandrini, Gye-Seon Lee, Florent Schaffhauser

Date Published: 2018

Publication Type: Journal

Domains of discontinuity in oriented flag manifolds

Abstract (Expand)

We study actions of discrete subgroups Γof semi-simple Lie groups G on associated oriented flag manifolds. These are quotients G/P, where the subgroup P lies between a parabolic subgroup and its identity … component. For Anosov subgroups Γ⊂G, we identify domains in oriented flag manifolds by removing a set obtained from the limit set of Γ, and give a combinatorial description of proper discontinuity and cocompactness of these domains. This generalizes analogous results of Kapovich-Leeb-Porti to the oriented setting. We give first examples of cocompact domains of discontinuity which are not lifts of domains in unoriented flag manifolds. These include in particular domains in oriented Grassmannians for Hitchin representations, which we also show to be nonempty. As a further application of the oriented setup, we give a new lower bound on the number of connected components of B-Anosov representations of a closed surface group into \SL(n,\mathbb R).

Authors: Florian Stecker, Nicolaus Treib

Date Published: 2018

Publication Type: Journal

Balanced ideals and domains of discontinuity of Anosov representations

Abstract (Expand)

We consider the action of Anosov subgroups of a semi-simple Lie group on the associated flag manifolds. A systematic approach to construct cocompact domains of discontinuity for this action was given … by Kapovich, Leeb and Porti in arXiv:1306.3837. For ∆-Anosov representations, we prove that every cocompact domain of discontinuity arises from this construction, up to a few exceptions in low rank. Then we compute which flag manifolds admit these domains and, in some cases, the number of domains. We also find a new compactification for locally symmetric spaces arising from maximal representations into \mathrmSp(4n+2,\mathbb R).

Author: Florian Stecker

Date Published: 2018

Publication Type: Journal

Pappus Theorem, Schwartz Representations and Anosov Representations

Abstract (Expand)

In the paper \textitPappus’s theorem and the modular group, R. Schwartz constructed a 2-dimensional family of faithful representations \rho_Θof the modular group \mathrmPSL(2,\mathbbZ) into the group … \mathscrG of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subgroup \mathrmPSL(2,\mathbbZ)_o of \mathrmPSL(2,\mathbbZ) under each representation \rho_Θis in the subgroup \mathrmPGL(3,\mathbbR) of \mathscrG and preserves a topological circle in the flag variety, but \rho_Θis \emphnot Anosov. In her PhD Thesis, V. P. Valério elucidated the Anosov-like feature of Schwartz representations: for every \rho_Θ, there exists a 1-dimensional family of Anosov representations ρ^\varepsilon_Θ of \mathrmPSL(2,\mathbbZ)_o into \mathrmPGL(3,\mathbbR) whose limit is the restriction of \rho_Θto \mathrmPSL(2,\mathbbZ)_o. In this paper, we improve her work: for each \rho_Θ, we build a 2-dimensional family of Anosov representations of \mathrmPSL(2,\mathbbZ)_o into \mathrmPGL(3,\mathbbR) containing ρ^\varepsilon_Θ and a 1-dimensional subfamily of which can extend to representations of \mathrmPSL(2,\mathbbZ) into \mathscrG. Schwartz representations are therefore, in a sense, the limits of Anosov representations of \mathrmPSL(2,\mathbbZ) into \mathscrG.

Authors: Thierry Barbot, Gye-Seon Lee, Viviane Pardini Valério

Date Published: 2018

Publication Type: Journal

Convex projective structures on nonhyperbolic three-manifolds

Abstract (Expand)

Y Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many submanifolds … each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist’s theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures.

Authors: Samuel Ballas, Jeffrey Danciger, Gye-Seon Lee

Date Published: 2018

Publication Type: Journal

Deformations of convex real projective manifolds and orbifolds

Abstract (Expand)

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with … an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric structures on orbifolds, and Hilbert geometry. The main examples of finitely generated groups for us will be Fuchsian groups, 3-manifold groups and Coxeter groups.

Authors: Suhyoung Choi, Gye-Seon Lee, Ludovic Marquis

Date Published: 2018

Publication Type: InCollection

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