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Currents, Systoles, and Compactifications of Character Varieties

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We study the Thurston–Parreau boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This … result is obtained as consequence of a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the intersection function on the set of closed curves is bi-Lipschitz equivalent to the length function with respect to a hyperbolic metric. The results of this paper on currents generalise the ones in arXiv:1710.07060v1 to the case of surfaces of finite area with geodesic boundary. Concerning the Thurston–Parreau boundary we improve on the results in arXiv:1710.07060v1 by showing that for higher rank groups, said open set of discontinuity is not empty. We give also explicit examples in the case of the \SL(3,\mathbb R)-Hitchin component.

Authors: M. Burger, A. Iozzi, A. Parreau, M. B. Pozzetti

Date Published: 2019

Publication Type: Misc

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

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We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact … bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In the appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichmüller translation distance.

Authors: Federico Franceschini, Maria Beatrice Pozzetti

Date Published: 2019

Publication Type: Book

Anti–de Sitter strictly GHC-regular groups which are not lattices

Abstract (Expand)

For d=4, 5, 6, 7, 8, we exhibit examples of \mathrm AdS^d,1 strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space \mathbbH^d, nor to any symmetric space. This provides a …s a negative answer to Question 5.2 in a work of Barbot et al. and disproves Conjecture 8.11 of Barbot-Mérigot [Groups Geom. Dyn. 6 (2012), pp. 441-483]. We construct those examples using the Tits representation of well-chosen Coxeter groups. On the way, we give an alternative proof of Moussong’s hyperbolicity criterion (Ph.D. Thesis) for Coxeter groups built on Danciger-Guéritaud-Kassel’s 2017 work and find examples of Coxeter groups W such that the space of strictly GHC-regular representations of W into \mathrm PO_d,2(\mathbbR) up to conjugation is disconnected.

Authors: Gye-Seon Lee, Ludovic Marquis

Date Published: 2019

Publication Type: Journal

Cross ratios on CAT(0) cube complexes and marked length-spectrum rigidity

Abstract (Expand)

We show that group actions on irreducible \CAT cube complexes with no free faces are uniquely determined by their \ell^1 length function. Actions are allowed to be non-proper and non-cocompact, as long … as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first length-spectrum rigidity result in a setting of \emphnon-positive curvature (with the exception of some particular cases in dimension 2 and symmetric spaces). As our main tool, we develop a notion of cross ratio on Roller boundaries of \CAT cube complexes. Inspired by results in \emphnegative curvature, we give a general framework reducing length-spectrum rigidity questions to the problem of extending cross-ratio preserving maps between (subsets of) Roller boundaries. The core of our work is then to show that, when there are no free faces, these cross-ratio preserving maps always extend to cubical isomorphisms. All our results equally apply to cube complexes with variable edge lengths. As a special case of our work, we construct a compactification of the Char\-ney–Stambaugh–Vogtmann Outer Space for the group of untwisted outer automorphisms of an (irreducible) right-angled Artin group. This generalises the length function compactification of the classical Culler–Vogtmann Outer Space.

Authors: Jonas Beyrer, Elia Fioravanti

Date Published: 2019

Publication Type: Misc

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

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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

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