The geometry of maximal representations of surface groups into SO(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.

arXiv comment56 pgs, section 3 has been reorganized , former sections 4.2 and 4.3 have been merged into section 4.2 and rewritten to avoid reference to maximal surfaces and Higgs bundles, appendix added on strong version of Ahlfors-Schwarz-Pick lemma. To appear in Duke Math Journal

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Cited by in corpus (6)

• Anosov representations with Lipschitz limit set
• Domination results in $n$-Fuchsian fibers in the moduli space of Higgs bundles
• Anosov representations acting on homogeneous spaces: domains of discontinuity
• Nearly geodesic immersions and domains of discontinuity
• Geometric structures for maximal representations and pencils
• Polygonal surfaces in pseudo-hyperbolic spaces