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81 Publications visible to you, out of a total of 81
Marked length spectrum rigidity for actions on CAT(0) cube complexes

Abstract (Expand)

Given an action on a metric space one can associate to each element of the group its translation length. This gives a function of the group to the reals called the marked length spectrum. Adding …ked length spectrum. Adding requirements for space and action, it is a natural question if the marked length spectrum already uniquely defines space and action. In this talk we want to show that this is the case when considering CAT(0) cube complexes (under some natural assumptions). The main tool to prove this will be a boundary rigidity using cross ratios. Joint workwith Elia Fioravanti.

Author: Jonas Beyrer

Date Published: 2019

Publication Type: Journal

Generalized stretch lines for surfaces with boundary

Abstract (Expand)

In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller … space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston’s stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodesics in the Teichmueller space of a closed/punctured surface that are not stretch lines in the sense of Thurston.

Authors: Daniele Alessandrini, Valentina Disarlo

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

Schottky groups and maximal representations

Abstract (Expand)

We describe a construction of Schottky type subgroups of automorphism groups of partially cyclically ordered sets. We apply this construction to the Shilov boundary of a Hermitian symmetric space and … show that in this setting Schottky subgroups correspond to maximal representations of fundamental groups of surfaces with boundary. As an application, we construct explicit fundamental domains for the action of maximal representations into \mathrm Sp(2n,\mathbb R) on \mathbb RP^2n-1.

Authors: Jean-Philippe Burelle, Nicolaus Treib

Date Published: 1st Aug 2018

Publication Type: Journal

Simultaneous ips on triangulated surfaces

Abstract (Expand)

We investigate a type of distance between triangulations on finite-type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism, … and our main results are upper bounds on the distance between triangulations that only depend on the topology of the surface.

Authors: Valentina Disarlo, Hugo Parlier

Date Published: 1st Aug 2018

Publication Type: Journal

Higgs bundles for real groups and the Hitchin-Kostant-Rallis section

Abstract (Expand)

We consider the moduli space of polystable-twisted-Higgs bundles over a compact Riemann surface, where is a real reductive Lie group and is a holomorphic line bundle over. Evaluating the Higgs field … on a basis of the ring of polynomial invariants of the isotropy representation defines the Hitchin map.

Authors: Oscar García-Prada, Ana Peón-Nieto, S. Ramanan

Date Published: 1st Apr 2018

Publication Type: Journal

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