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62 Publications visible to you, out of a total of 62
Domains of discontinuity in oriented flag manifolds

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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

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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

A complete description of the antipodal set of most symmetric spaces of compact type

Abstract (Expand)

It is known that the antipodal set of a Riemannian symmetric space of compact type G/K consists of a union of K-orbits. We determine the dimensions of these K-orbits of most irreducible symmetric spaces … of compact type. The symmetric spaces we are not going to deal with are those with restricted root system \mathfrak a_r and a non-trivial fundamental group, which is not isomorphic to \mathbb Z_2 or \mathbb Z_r+1. For example, we show that the antipodal sets of the Lie groups Spin(2r+1)r≥5, E_8 and G_2 consist only of one orbit which is of dimension 2r, 128 and 6, respectively; \SO(2r+1) has also an antipodal set of dimension 2r; and the Grassmannian Gr_r,r+q(R) has a rq-dimensional orbit as antipodal set if r≥5 and r≠q>0.

Author: Jonas Beyrer

Date Published: 2018

Publication Type: Journal

Cubulations and cross ratios on contracting boundaries

Abstract (Expand)

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations … and geodesic currents. We add to this picture by studying cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary \partial_∞G. A consequence of our results is that essential, hyperplane-essential cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work in arXiv:1903.02447. Along the way, we describe the relationship between the Roller boundary of a \mathrmCAT(0) cube complex, its Gromov boundary and - in the non-hyperbolic case - the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

Authors: Jonas Beyrer, Elia Fioravanti

Date Published: 2018

Publication Type: Journal

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