## Duke Mathematical Journal

### The geometry of maximal representations of surface groups into $\mathrm{SO}_{0}(2,n)$

#### 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
Revised: 10 April 2019
First available in Project Euclid: 30 September 2019

https://projecteuclid.org/euclid.dmj/1569830546

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

Mathematical Reviews number (MathSciNet)
MR4017517

#### 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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