The geometry of maximal representations of surface groups into $\mathrm{SO}_0(2,n)$

Brian Collier (1) , Nicolas Tholozan (2, 3) , Jérémy Toulisse (4)

1 Department of Mathematics [Maryland]
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 DMA - Département de Mathématiques et Applications - ENS Paris
4 USC - University of Southern California

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 Fuch-sian 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.

Domaines: Mathématiques [math]; Géométrie différentielle [math.DG]; Topologie géométrique [math.GT]

Brian Collier, Nicolas Tholozan, Jérémy Toulisse. The geometry of maximal representations of surface groups into $\mathrm{SO}_0(2,n)$. Duke Mathematical Journal, 2019, 168 (15), pp.2873-2949. ⟨10.1215/00127094-2019-0052⟩. ⟨hal-01792650⟩