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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 describethe latter by constructing a shear cocycle from the cross ratio naturallyassociated to $\rho$. The process developed to this purpose applies to a wideclass of cross ratios, including examples arising from Hitchin and$\Theta$-positive representations in $\mathrm{SO}(p,q)$. We also show that thelength spectrum of $\rho$ dominates the ones of $\rho$-invariant pleatedsurfaces, with strict inequality exactly on curves that intersect the bendinglocus.  We observe that the canonical decomposition of a $\rho$-invariant pleatedsurface into leaves and plaques corresponds to a decomposition of theGuichard-Wienhard domain of discontinuity of $\rho$ into standard fiberedblocks, namely triangles and lines of photons. Conversely, we give a concreteconstruction of photon manifolds fibering over hyperbolic surfaces by gluingtogether 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 thecorrespondence between maximal representations and fibered photon manifoldsthrough a constructive and geometric approach, bypassing the use of Higgsbundles.

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