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

Let $S$ be a closed surface of genus $g\ge 2$. We study the cocompact domain of discontinuity ${\mathrm{\Omega }}_{\rho }$ in the Einstein universe ${\mathrm{E}\mathrm{i}\mathrm{n}}^{p-1,p}$ defined by Guichard-Wienhard and Kapovich-Leeb-Porti for a class of $p$-Anosov representations $\rho :{\pi }_{1}S\to {\mathrm{S}\mathrm{O}}_0(p,p+1)$ including Hitchin representations, for $p\ge 3$. The quotient $M_{\rho }=\rho ({\pi }_{1}S)\mathrm{\backslash }{\mathrm{\Omega }}_{\rho }$ 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 ${\mathfrak{F}}_{\rho }\to M_{\rho }\to S$ is trivial.