Collar lemma for Hitchin representations

Lee, Gye-Seon; Zhang, Tengren

doi:10.2140/gt.2017.21.2243

Cited by 17 publications

(11 citation statements)

References 27 publications

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“…However it is worth remarking that, as opposed to the classical Collar Lemma, Theorem 1.8 is not just a consequence of the Margulis Lemma: in our setting the sets of minimal displacement of the isometries ρ(γ) and ρ(η) do not necessarily intersect. A similar version of the Collar Lemma in the framework of Hitchin representations has been recently established in [LZ14] (see Remark 3.5 for a comparison with our results).…”

Section: Introductionsupporting

confidence: 82%

Maximal representations, non-Archimedean Siegel spaces, and buildings

Burger

Pozzetti

2017

Geom. Topol.

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Abstract. Let F be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in Sp(2n, F). We show that ultralimits of maximal representations in Sp(2n, R) admit such a framing, and that all maximal framed representations satisfy a suitable generalization of the classical Collar Lemma. In particular this establishes a Collar Lemma for all maximal representations into Sp(2n, R). We then describe a procedure to get from representations in Sp(2n, F) interesting actions on affine buildings, and, in the case of representations admitting a maximal framing, we describe the structure of the elements of the group acting with zero translation length.

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Section: Introductionsupporting

confidence: 82%

Maximal representations, non-Archimedean Siegel spaces, and buildings

Burger

Pozzetti

2017

Geom. Topol.

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“…It roughly says that, if γ and η are two essentially intersecting curves on Σ, then L ρ (γ) and L ρ (η) cannot both be small. Such a collar lemma was obtained by Lee and Zhang for Hitchin representations into SL(n, R) [LZ17] and by Burger and Pozzetti [BP17] for maximal representations into Sp(2n, R). More precisely, they prove: Theorem 1.7.…”

Section: Introductionmentioning

confidence: 77%

The geometry of maximal representations of surface groups into SO0(2,n)

Collier¹,

Tholozan²,

Toulisse³

2019

Duke Math. J.

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In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest.We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.

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“…Following the work of N. Hitchin, it became apparent that the spaces identified, now called Hitchin components, include representations with important geometric features. For instance, F. Labourie introduced in [57] the notion of an Anosov representation and used techniques from dynamical systems to prove (among other essential geometric properties) that representations lying inside the component of Hitchin for G = PSL (n, R), PSp (2n, R) or PO (n, n + 1) are faithful with discrete image; we refer the reader to [11], [37], [38], [58], [59], [62], [73] for subsequent works on the geometric and dynamical properties of representations in the Hitchin components.…”

Section: Higher Teichmüller Theorymentioning

confidence: 99%

From hyperbolic Dehn filling to surgeries in representation varieties

Kydonakis¹

2021

Preprint

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In this semi-expository article we describe a gluing method developed for constructing certain model objects in representation varieties Hom (π1 (Σ) , G) for a topological surface Σ and a semisimple Lie group G. Explicit examples are demonstrated in the case of Θ-positive representations lying in the p•(2g − 2)−1 many exceptional connected components of the SO (p, p + 1)-character variety for p > 2.

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Collar lemma for Hitchin representations

Abstract: Abstract. In this article, we prove an analog of the classical collar lemma in the setting of Hitchin representations.

Cited by 17 publications

References 27 publications

Maximal representations, non-Archimedean Siegel spaces, and buildings

Maximal representations, non-Archimedean Siegel spaces, and buildings

The geometry of maximal representations of surface groups into SO0(2,n)

From hyperbolic Dehn filling to surgeries in representation varieties

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