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Affine Anosov representations and proper actions

Abstract (Expand)

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group \SO^0(n+1,n)⋉\bR^2n+1. We then show that a representation ρof a word hyperbolic group is affine … Anosov if and only if its linear part \mathttL_ρis Anosov in \mathsfSO^0(n+1,n) with respect to the stabilizer of a maximal isotropic plane and ρ(Γ) acts properly on \mathbbR^2n+1.

Authors: Sourav Ghosh, Nicolaus Treib

Date Published: 2017

Publication Type: Misc

On order-preserving representations

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In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal … representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermitian type are faithful with discrete image and that the set of order preserving representations is closed in the representation variety. For Lie groups of Hermitian type whose associated symmetric space is of tube type we give a geometric characterization of these representations in terms of the causal structure on the Shilov boundary.

Authors: G. Ben Simon, M. Burger, T. Hartnick, A. Iozzi, A. Wienhard

Date Published: 29th Jun 2016

Publication Type: Journal

Polyhedral Horofunction Compactification as Polyhedral Ball

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In this paper we answer positively a question raised by Kapovich and Leeb in a recent paper titled "Finsler bordifications of symmetric and certain locally symmetric spaces". Specifically, … we show that for a finite-dimensional vector space with polyhedral norm, its horofunction compactification is homeomorphic to the dual unit ball of the norm by an explicit map. To prove this we establish a criterion for converging sequences in the horofunction compactification, and generalize the basic notion of the moment map in the theory of toric varieties.

Authors: Lizhen Ji, Anna-Sofie Schilling

Date Published: 2016

Publication Type: Misc

Representations and geometric structures

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This note summarizes in an informal way some geometric properties of Anosov representations into the symplectic group, which were presented in a talk at the conference What is Next. The mathematical … legacy of Bill Thurston, held in June 2014 in Cornell.

Author: Anna Wienhard

Date Published: 2016

Publication Type: Misc

Basmajian-type inequalities for maximal representations

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For suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize … the classical Basmajian’s identity from Teichmueller theory. Any equality characterizes diagonal embeddings.

Authors: Federica Fanoni, Maria Beatrice Pozzetti

Date Published: 2016

Publication Type: Misc

Holonomy map fibers of ℂP1-structures in moduli space

Abstract (Expand)

Let S be a closed oriented surface of genus g≥2. Fix an arbitrary non-elementary representation ρ\col\pi_1(S)\to \rm SL_2(\C) and consider all marked (complex) projective structures on S with holonomy …my ρ. We show that their underlying conformal structures are dense in the moduli space of S.

Authors: Shinpei Baba, Subhojoy Gupta

Date Published: 31st Aug 2015

Publication Type: Journal

Cameral data for SU(p+1,p)-Higgs bundles

Abstract (Expand)

We study the cameral and spectral data for the moduli space of polystable \SU(p+1,p)-Higgs bundles and deduce the latter from the former. As an application, we obtain that the Toledo invariant classifies … the connected components of the regular fibers of the Hitchin map.

Author: Ana Peón-Nieto

Date Published: 2015

Publication Type: Misc

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