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Cross ratios, Anosov representations and the energy functional on Teichmüller space

François Labourie (2008)

Annales scientifiques de l'École Normale Supérieure

We study two classes of linear representations of a surface group: Hitchin and maximal symplectic representations. We relate them to cross ratios and thus deduce that they are displacing which means that their translation lengths are roughly controlled by the translations lengths on the Cayley graph. As a consequence, we show that the mapping class group acts properly on the space of representations and that the energy functional associated to such a representation is proper. This implies the existence...

Cross ratios, surface groups, P S L ( n , 𝐑 ) and diffeomorphisms of the circle

François Labourie (2007)

Publications Mathématiques de l'IHÉS

This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into PSL(n,ℝ) – known as the n-Hitchin component– to a subset of the set of cross ratios on the boundary at infinity of the group. Similarly, we study some representations into C 1 , h ( 𝕋 ) ⋊ Diff h ( 𝕋 ) associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains alln-Hitchin components as well as the set of...

Déformation localisée de surfaces de Riemann.

Peter Haïssinsky (2005)

Publicacions Matemàtiques

Let Y be a Riemann surface with compact boundary embedded into a hyperbolic Riemann surface of finite type X. It is proved that the space of deformations D of Y into X is an open subset of the Teichmüller space T(X) of X. Furthermore, D has compact closure if and only if Y is simply connected or isomorphic to a punctured disk, and D= T(X) if and only if the components of X Y are all disks or punctured disks.

Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call acluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space Ł Γ of line bundles with connections on the graph Γ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ 1 and Γ 2 areequivalentif the Newton polygons of the corresponding partition functions...

Diophantine approximation on Veech surfaces

Pascal Hubert, Thomas A. Schmidt (2012)

Bulletin de la Société Mathématique de France

We show that Y. Cheung’s general Z -continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental...

Ergodic theory of interval exchange maps.

Marcelo Viana (2006)

Revista Matemática Complutense

A unified introduction to the dynamics of interval exchange maps and related topics, such as the geometry of translation surfaces, renormalization operators, and Teichmüller flows, starting from the basic definitions and culminating with the proof that almost every interval exchange map is uniquely ergodic. Great emphasis is put on examples and geometric interpretations of the main ideas.

Geometric characterization for affine mappings and Teichmüller mappings

Zhiguo Chen (2003)

Studia Mathematica

We characterize affine mappings on the unit disk and on rectangles by module conditions. The main result generalizes the classic Schwarz lemma. As an application, we give a sufficient condition for a K-quasiconformal mapping on a Riemann surface to be a Teichmüller mapping.

Hyperbolic spaces in Teichmüller spaces

Christopher J. Leininger, Saul Schleimer (2014)

Journal of the European Mathematical Society

We prove, for any n , that there is a closed connected orientable surface S so that the hyperbolic space ℍ n almost-isometrically embeds into the Teichmüller space of S , with quasi-convex image lying in the thick part. As a consequence, ℍ n quasi-isometrically embeds in the curve complex of S .

Invariants of translation surfaces

Pascal Hubert, Thomas A. Schmidt (2001)

Annales de l’institut Fourier

We definite invariants of translation surfaces which refine Veech groups. These aid in exact determination of Veech groups. We give examples where two surfaces of isomorphic Veech group cannot even share a common tree of balanced affine coverings. We also show that there exist translation surfaces of isomorphic Veech groups which cannot affinely cover any common surface. We also extend a result of Gutkin and Judge and thereby give the first examples of noncompact Fuchsian...

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