Positively ratioed representations

Martone, Giuseppe; Zhang, Tengren

doi:10.48550/arxiv.1609.01245

Cited by 3 publications

(6 citation statements)

References 20 publications

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“…Some Anosov representations of surface groups, such as Hitchin representations into real split Lie groups or maximal representations into Hermitian Lie groups, have the additional property of forming connected components of the whole space of representations. There have been several attempts to propose a unifying characterization of these representations (see [MZ16] and [GW16]). Note that quasi-Fuchsian representations into PSL(2, C) do not form components; indeed, they can be continuously deformed into representations with non-discrete image.…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

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

Collier¹,

Tholozan²,

Toulisse³

2019

Duke Math. J.

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“…Labourie [Lab07] has given one of the cross ratios in Example 3.13 ad-hoc and used it as tool to understand Hitchin representations. Moreover, Martone and Zhang [MZ17] have constructed cross ratios on boundaries of surface groups, which in particular for SL(n, R)-Hitchin representations coincide with the pullback under the boundary map of some of the cross ratios in Example 3.13.…”

Section: Related Workmentioning

confidence: 99%

“…We remark that some specific of those cross ratios are known already and have been used for analyzing Hitchin representations (see e.g. [Lab07], [MZ17]).…”

Section: We Remind That Any Isometrymentioning

confidence: 99%

“…On the boundary ∂ ∞ S of the universal cover of a closed surface S there are many other cross ratios, besides the above constructed one, that parametrize classical objects associated to the surface; such as simple closed curves, measured laminations, points of Teichmüller space [Bon88], Hitchin representations [Lab07] and positively ratioed representations [MZ17] 1 -to name a few. This prominence and importance of cross ratios in negative curvature motivates us to ask if such objects also exists for non-positively curved spaces and how much information about the geometry they carry.…”

Section: Introductionmentioning

confidence: 99%

“…The natural boundary maps Anosov representations come with can be used to pullback the cross ratios that we construct to get cross ratios on the boundary of the group. This has been done for representation into SL(n, R) in [Lab07] using an ad-hoc definition of a cross ratio and this is in the spirit of [MZ17]; and we hope that the vector valued cross ratio we analyze here allows for further applications in this area.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cross Ratios on Boundaries of Symmetric Spaces and Euclidean Buildings

Beyrer

2020

Transformation Groups

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We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even CAT(−1) space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings -we obtain vector valued cross ratios defined on simplices of the building at infinity. We show several properties of those cross ratios; for example that (under some restrictions) periods of hyperbolic isometries give back the translation vector. In addition, we show that cross ratio preserving maps on the chamber set are induced by isometries and vice versa -motivating that the cross ratios bring the geometry of the symmetric space/Euclidean building to the boundary.

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An invitation to higher Teichmüller theory

Wienhard¹

2018

Preprint

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Positively ratioed representations

Cited by 3 publications

References 20 publications

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

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

Cross Ratios on Boundaries of Symmetric Spaces and Euclidean Buildings

An invitation to higher Teichmüller theory

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