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62 Publications visible to you, out of a total of 62
Horofunction Compactifications of Symmetric Spaces

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We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction … compactification with respect to a polyhedral Finsler metric.

Authors: Thomas Haettel, Anna-Sofie Schilling, Cormac Walsh, Anna Wienhard

Date Published: 2017

Publication Type: Misc

Anosov representations and proper actions

Abstract (Expand)

We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using … a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups.

Authors: François Guéritaud, Olivier Guichard, Fanny Kassel, Anna Wienhard

Date Published: 2017

Publication Type: Journal

Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

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We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group G. These manifolds … are obtained as quotients of the domains of discontinuity in generalized flag varieties G/P constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For G-Fuchsian representations and their Anosov deformations, where G is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For G-Fuchsian and G-quasi-Fuchsian representations in simple G of rank at least two, we show that the quotient manifolds are not Kähler. We also describe the Picard groups of these quotient manifolds, compute the cohomology of line bundles on them, and show that for G of sufficiently large rank these manifolds admit nonconstant meromorphic functions. In a final section, we apply our topological results to several explicit families of domains and derive closed formulas for topological invariants in some cases. We also show that the quotient manifold for a G-Fuchsian representation in \mathrmPSL_3(C) is a fiber bundle over a surface, and we conjecture that this holds for all simple G.

Authors: David Dumas, Andrew Sanders

Date Published: 2017

Publication Type: Misc

Bi-Lagrangian structures and Teichmüller theory

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This paper has two purposes: the first is to study several structures on manifolds in the general setting of real and complex differential geometry; the second is to apply this study to Teichmüller … theory. We primarily focus on bi-Lagrangian structures, which are the data of a symplectic structure and a pair of transverse Lagrangian foliations, and are equivalent to para-Kähler structures. First we carefully study real and complex bi-Lagrangian structures and discuss other closely related structures and their interrelationships. Next we prove the existence of a canonical complex bi-Lagrangian structure in the complexification of any real-analytic Kähler manifold and showcase its properties. We later use this bi-Lagrangian structure to construct a natural almost hyper-Hermitian structure. We then specialize our study to moduli spaces of geometric structures on closed surfaces, which tend to have a rich symplectic structure. We show that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory, while revealing other new geometric features. We also gain clarity on several well-known results of Teichmüller theory by deriving them from pure differential geometric machinery.

Authors: Brice Loustau, Andrew Sanders

Date Published: 2017

Publication Type: Misc

Flows on the PGL(V)-Hitchin component

Abstract (Expand)

In this article we define new flows on the Hitchin components for \mathrmPGL(V). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. … Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when n = 2. Using these flows, we construct a global coordinate system on the Hitchin component. In a companion paper to this article two of the authors develop new tools to compute the Goldman symplectic form on the Hitchin component, and prove that this global coordinate system is a Darboux coordinate system.

Authors: Zhe Sun, Anna Wienhard, Tengren Zhang

Date Published: 2017

Publication Type: Misc

A structure theorem for geodesic currents and length spectrum compactifications

Abstract (Expand)

We find a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either … is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the intersection function on the set of closed curves is bilipschitz equivalent to the length function with respect to a hyperbolic metric. We show that the subset of currents with positive systole is open and that the mapping class group acts properly discontinuously on it. As an application, we obtain in the case of compact surfaces a structure theorem on the length functions appearing in the length spectrum compactification both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group.

Authors: Marc Burger, Alessandra Iozzi, Anne Parreau, Maria Beatrice Pozzetti

Date Published: 2017

Publication Type: Misc

Collar lemma for Hitchin representations

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There is a classical result known as the collar lemma for hyperbolic surfaces. A consequence of the collar lemma is that if two closed curves A and B on a closed orientable hyperbolizable surface … intersect each other, then there is an explicit lower bound for the length of A in terms of the length of B, which holds for every hyperbolic structure on the surface. In this article, we prove an analog of the classical collar lemma in the setting of Hitchin representations.

Authors: Gye-Seon Lee, Tengren Zhang

Date Published: 2017

Publication Type: Journal

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