Duke Mathematical Journal

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

Brian Collier, Nicolas Tholozan, and Jérémy Toulisse

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Abstract

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.

Article information

Source
Duke Math. J., Volume 168, Number 15 (2019), 2873-2949.

Dates
Received: 14 June 2017
Revised: 10 April 2019
First available in Project Euclid: 30 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1569830546

Digital Object Identifier
doi:10.1215/00127094-2019-0052

Mathematical Reviews number (MathSciNet)
MR4017517

Subjects
Primary: 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]
Secondary: 53C50: Lorentz manifolds, manifolds with indefinite metrics 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05] 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 58E12: Applications to minimal surfaces (problems in two independent variables) [See also 49Q05] 57M50: Geometric structures on low-dimensional manifolds

Keywords
maximal representations Higgs bundles pseudo-Riemannian geometry Anosov representations geometric structures

Citation

Collier, Brian; Tholozan, Nicolas; Toulisse, Jérémy. The geometry of maximal representations of surface groups into $\mathrm{SO}_{0}(2,n)$. Duke Math. J. 168 (2019), no. 15, 2873--2949. doi:10.1215/00127094-2019-0052. https://projecteuclid.org/euclid.dmj/1569830546


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