On Einstein Structures for $\mathrm{SO}_0(p,p+1)$-Surface Group Representations

Let $S$ be a closed surface of genus $g \geq 2$. We study the cocompact domain of discontinuity $\Omega_{\rho}$ in the Einstein universe $\mathrm{Ein}^{p-1,p}$ defined by Guichard-Wienhard and Kapovich-Leeb-Porti for a class of $p$-Anosov representations $\rho:\pi_1S \rightarrow \mathrm{SO}_0(p,p+1)$ including Hitchin representations, for $p \geq 3$. The quotient $M_{\rho} = \rho(\pi_1S)\backslash \Omega_{\rho}$ is abstractly known to be realizable as a fiber bundle over $S$, with unknown fiber of unique homotopy type $F_{\rho}$. We explicitly exhibit $M_{\rho}$ as a smooth $\mathfrak{F}_{\rho}$-fiber bundle over $S$, determining the diffeomorphism type of $M_{\rho}$ and the unique homotopy type $F_{\rho}$. Surprisingly, in many situations the fiber bundle $\mathfrak{F}_{\rho} \rightarrow M_{\rho}\rightarrow S$ is trivial.