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

Abstract

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

Substantial changes from v1, including title change. We now address the global topology for all iota-Fuchsian deformations. 35 pages, 4 figures, 1 table