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81 Publications visible to you, out of a total of 81
The pre-symplectic geometry of opers and the holonomy map

Abstract (Expand)

Given a connected complex semi-simple Lie group G and a Riemann surface X, a G-oper on X is a higher rank generalization of a complex projective structure on X. These objects play an important role in …in integrable systems and geometric representation theory, a status that was cemented by the seminal work of Beilinson-Drinfeld \citeBD91. For G a connected complex simple Lie group of adjoint type, we study the global deformation theory of G-opers on a connected, closed, oriented smooth surface Σof genus at least two. We exhibit the deformation space of G-opers on Σas a holomorphic fiber bundle over Teichmüller space, and elucidate the relationship with the deformation space of complex projective structures. Then, we show that there is a family of identifications of the deformation space of G-opers with a holomorphic vector bundle \mathcalB_G(Σ) over Teichmüller space whose typical fiber over a Riemann surface X is a sum of spaces of pluri-canonical sections. Finally, we show that the holonomy map from the deformation space of G-opers to the deformation space of flat G-bundles on Σis a holomorphic immersion. As a consequence of this result, we show that the deformation space of G-opers carries a (pre-symplectic) closed holomorphic differential 2-form of constant rank, and we prove that a sub-family of the identifications of \mathcalB_G(Σ) with the deformation space of G-opers is a holomorphic pre-symplectic map for a natural holomorphic pre-symplectic form on \mathcalB_G(Σ). These results generalize the fundamental features of the deformation space of complex projective structures on Σto the setting of G-opers.

Author: Andrew Sanders

Date Published: 2018

Publication Type: Journal

An invitation to higher Teichmüller theory

Abstract

The goal of this article is to invite the reader to get to know and to get involved into higher Teichmüller theory by describing some of its many facets.

Author: Anna Wienhard

Date Published: 2018

Publication Type: Journal

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

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