$\mathrm{SO}_0(2,n+1)$-maximal representations and hyperbolic surfaces

TL;DR

This paper introduces a geometric framework for studying maximal representations of surface groups into $ ext{SO}_0(2,n+1)$, using $ ho$-invariant pleated surfaces in $ ext{H}^{2,n}$ to analyze their properties and associated structures.

Contribution

It develops a new geometric approach to analyze maximal representations via $ ho$-invariant pleated surfaces, connecting them to fibered photon manifolds and extending previous results.

Findings

$ ho$-invariant pleated surfaces are embedded, acausal, and have a hyperbolic structure.

Constructs shear cocycles from cross ratios associated to $ ho$.

Length spectrum of $ ho$ dominates that of pleated surfaces.

Abstract

We study maximal representations of surface groups $\rho:\pi_1(\Sigma)\to\mathrm{SO}_0(2,n+1)$ via the introduction of $\rho$-invariant pleated surfaces inside the pseudo-Riemannian space $\mathbb{H}^{2,n}$ associated to maximal geodesic laminations of $\Sigma$. We prove that $\rho$-invariant pleated surfaces are always embedded, acausal, and possess an intrinsic pseudo-metric and a hyperbolic structure. We describe the latter by constructing a shear cocycle from the cross ratio naturally associated to $\rho$. The process developed to this purpose applies to a wide class of cross ratios, including examples arising from Hitchin and $\Theta$-positive representations in $\mathrm{SO}(p,q)$. We also show that the length spectrum of $\rho$ dominates the ones of $\rho$-invariant pleated surfaces, with strict inequality exactly on curves that intersect the bending locus. We observe that the canonical decomposition of a $\rho$-invariant pleated surface into leaves and plaques corresponds to a decomposition of the Guichard-Wienhard domain of discontinuity of $\rho$ into standard fibered blocks, namely triangles and lines of photons. Conversely, we give a concrete construction of photon manifolds fibering over hyperbolic surfaces by gluing together triangles of photons. The tools we develop allow to recover various results by Collier, Tholozan, and Toulisse on the (pseudo-Riemannian) geometry of $\rho$ and on the correspondence between maximal representations and fibered photon manifolds through a constructive and geometric approach, bypassing the use of Higgs bundles.