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62 Publications visible to you, out of a total of 62
Transgression in bounded cohomology and a conjecture of Monod

Abstract (Expand)

We develop an algebro-analytic framework for the systematic study of the con-tinuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology … of \mathrmPSL(2,\mathbb R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles.

Author: Andreas Ott

Date Published: 15th Nov 2019

Publication Type: Journal

Surface groups acting on CAT (-1) spaces

Abstract (Expand)

Harmonic map theory is used to show that a convex cocompact surface group action on a \mathrmCAT(-1) metric space fixes a convex copy of the hyperbolic plane (i.e. the action is Fuchsian) if and only … if the Hausdorff dimension of the limit set of the action is equal to 1. This provides another proof of a result of Bonk and Kleiner. More generally, we show that the limit set of every convex cocompact surface group action on a \mathrmCAT(-1) space has Hausdorff dimension ≥1, where the inequality is strict unless the action is Fuchsian.

Authors: GEORGIOS DASKALOPOULOS, CHIKAKO MESE, ANDREW SANDERS, ALINA VDOVINA

Date Published: 1st Jul 2019

Publication Type: Journal

Higgs Bundles and Geometric Structures on Manifolds

Abstract (Expand)

Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichmüller … theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin’s equations. Baraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to construct geometric structures in some interesting cases. In this paper, we will explain the main ideas behind this theory and we will survey some recent results in this direction, which are joint work with Qiongling Li.

Author: Daniele Alessandrini

Date Published: 10th May 2019

Publication Type: Journal

(G,P)-opers and global Slodowy slices

Abstract (Expand)

In this paper, we introduce a generalization of G-opers for arbitrary parabolic subgroups P<G. For parabolic subgroups associated to even nilpotents, we parameterize (G,P)-opers by an object … generalizing the base of the Hitchin fibration. In particular, we describe families of opers associated to higher Teichmuller spaces

Authors: Brian Collier, Andrew Sanders

Date Published: 2019

Publication Type: Misc

Noncommutative coordinates for symplectic representations

Abstract (Expand)

We introduce coordinates on the space of Lagrangian decorated and framed representations of the fundamental group of a surface with punctures into the symplectic group Sp(2n,R). These coordinates provide … a non-commutative generalization of the parametrizations of the spaces of representations into \mathrmSL(2,R) given by Thurston, Penner, and Fock-Goncharov. With these coordinates, the space of framed symplectic representations provides a geometric realization of the non-commutative cluster algebras introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of decorated maximal representations. We use this to determine the homotopy type of the space of decorated maximal representations, and its homeomorphism type when n=2.

Authors: Daniele Alessandrini, Olivier Guichard, Evgenii Rogozinnikov, Anna Wienhard

Date Published: 2019

Publication Type: Misc

Anosov representations with Lipschitz limit set

Abstract (Expand)

We study Anosov representation for which the image of the boundary map is the graph of a Lipschitz function, and show that the orbit growth rate with respect to an explicit linear function, the unstable … Jacobian, is integral. Several applications to the orbit growth rate in the symmetric space are provided.

Authors: Beatrice Pozzetti, Andrés Sambarino, Anna Wienhard

Date Published: 2019

Publication Type: Misc

Conformality for a robust class of non-conformal attractors

Abstract (Expand)

In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence … property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent. In the appendix, in collaboration with M. Bridgeman, we extend a classical result on the Hessian of the Hausdorff dimension on purely imaginary directions.

Authors: Beatrice Pozzetti, Andrés Sambarino, Anna Wienhard

Date Published: 2019

Publication Type: Misc

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