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81 Publications visible to you, out of a total of 81
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

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

Abstract (Expand)

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

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