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Flows on the $$\mathbf{PGL(V)}$$-Hitchin Component

Abstract (Expand)

In this article we define new flows on the Hitchin components for PSL(n,R). 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. We determine a global coordinate system on the Hitchin component. Using the computation of the Goldman symplectic form on the Hitchin component, that is developed by two of the authors in a companion paper to this article (Sun and Zhang in The Goldman symplectic form on the PGL(V)-Hitchin component, 2017. arXiv:1709.03589), this gives a global Darboux coordinate system on the Hitchin component.

Authors: Zhe Sun, Anna Wienhard, Tengren Zhang

Date Published: 1st Apr 2020

Publication Type: Journal

Topological mixing of Weyl chamber flows

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In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological … mixing.

Authors: NGUYEN-THI DANG, OLIVIER GLORIEUX

Date Published: 17th Feb 2020

Publication Type: Journal

Renormalizing an infinite rational IET

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We study an interval exchange transformation of [0,1] formed by cutting the interval at the points 1/n and reversing the order of the intervals. We find that the transformation is periodic away from a … Cantor set of Hausdorff dimension zero. On the Cantor set, the dynamics are nearly conjugate to the 2-adic odometer.

Authors: W. Patrick Hooper, Kasra Rafi, Anja Randecker

Date Published: 2020

Publication Type: Journal

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

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