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19 Publications visible to you, out of a total of 19
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

Limits of Geometries

Abstract (Expand)

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e., the homogeneous space on which the structures are modeled, abruptly changes. … In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spaces, describing the basic process by which one homogeneous geometry may transform into another. We develop a general framework to describe transitions in the context that both geometries involved are represented as sub-geometries of a larger ambient geometry. Specializing to the setting of real projective geometry, we classify the geometric limits of any sub-geometry whose structure group is a symmetric subgroup of the projective general linear group. As an application, we classify all limits of three-dimensional hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and Sol geometry among the limits. We prove, however, that the other Thurston geometries, in particular \mathbbH^2 \times \mathbbR and \widetilde \textup SL_2\mathbbR, do not embed in any limit of hyperbolic geometry in this sense.

Authors: Daryl Cooper, Jeffrey Danciger, Anna Wienhard

Date Published: 1st Sep 2018

Publication Type: Journal

Positivity and higher Teichmüller theory

Abstract (Expand)

We introduce Θ-positivity, a new notion of positivity in real semisimple Lie groups. The notion of Θ-positivity generalizes at the same time Lusztig’s total positivity in split real Lie groups as well … as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, \SO(p,q) for p<q, and a family of exceptional Lie groups, which admit a Θ-positive structure. We describe key aspects of Θ-positivity and make a connection with representations of surface groups and higher Teichmüller theory.

Authors: Olivier Guichard, Anna Wienhard

Date Published: 6th Aug 2018

Publication Type: InCollection

Deforming convex real projective structures

Abstract (Expand)

Let S be a closed, connected, orientable surface of genus at least 2, and let C(S) denote the deformation space of convex real projective structures S. In this article, we introduce two new flows on … C(S), which we call the internal bulging flow and the eruption flow. These are geometrically defined flows associated to each pair of pants in a pants decomposition on S that deform the internal parameters. We show that the eruption flows, together with the generalized twist flows about the pants curves, give rise to a half-dimensional family of commuting flows on C(S).

Authors: Anna Wienhard, Tengren Zhang

Date Published: 1st Feb 2018

Publication Type: Journal

An invitation to higher Teichmüller theory

Abstract

The goal of this article is to invite the reader to get to know and to get involved into higher Teichmüller theory by describing some of its many facets.

Author: Anna Wienhard

Date Published: 2018

Publication Type: Journal

On weakly maximal representations of surface groups

Abstract (Expand)

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. We prove that weakly maximal representations are discrete … and injective and we describe the structure of the Zariski closure of their image. Furthermore, we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representations variety.

Authors: Gabi Ben Simon, Marc Burger, Tobias Hartnick, Alessandra Iozzi, Anna Wienhard

Date Published: 1st Mar 2017

Publication Type: Journal

Horofunction Compactifications of Symmetric Spaces

Abstract (Expand)

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

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