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81 Publications visible to you, out of a total of 81
Bi-Lagrangian structures and Teichmüller theory

Abstract (Expand)

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

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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

Abstract (Expand)

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

Affine Anosov representations and proper actions

Abstract (Expand)

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group \SO^0(n+1,n)⋉\bR^2n+1. We then show that a representation ρof a word hyperbolic group is affine … Anosov if and only if its linear part \mathttL_ρis Anosov in \mathsfSO^0(n+1,n) with respect to the stabilizer of a maximal isotropic plane and ρ(Γ) acts properly on \mathbbR^2n+1.

Authors: Sourav Ghosh, Nicolaus Treib

Date Published: 2017

Publication Type: Misc

On order-preserving representations

Abstract (Expand)

In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal … representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermitian type are faithful with discrete image and that the set of order preserving representations is closed in the representation variety. For Lie groups of Hermitian type whose associated symmetric space is of tube type we give a geometric characterization of these representations in terms of the causal structure on the Shilov boundary.

Authors: G. Ben Simon, M. Burger, T. Hartnick, A. Iozzi, A. Wienhard

Date Published: 29th Jun 2016

Publication Type: Journal

Polyhedral Horofunction Compactification as Polyhedral Ball

Abstract (Expand)

In this paper we answer positively a question raised by Kapovich and Leeb in a recent paper titled "Finsler bordifications of symmetric and certain locally symmetric spaces". Specifically, … we show that for a finite-dimensional vector space with polyhedral norm, its horofunction compactification is homeomorphic to the dual unit ball of the norm by an explicit map. To prove this we establish a criterion for converging sequences in the horofunction compactification, and generalize the basic notion of the moment map in the theory of toric varieties.

Authors: Lizhen Ji, Anna-Sofie Schilling

Date Published: 2016

Publication Type: Misc

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