Projective deformations of hyperbolic 3-orbifolds with turnover ends
Alejandro García, Joan Porti
TL;DR
The paper investigates projective deformations of topologically finite hyperbolic 3-orbifolds whose ends have turnover cross sections, proving that any deformation yields a properly convex projective structure with ends that are either hyperbolic cusps or totally geodesic. It develops a local model around the turnover cusp $[0,\infty)\times S^2(3,3,3)$ by constructing a 2-dimensional slice near the hyperbolic holonomy $\rho_{\mathrm{hyp}}$, then extends these local deformations to the global setting using a holonomy principle to show that generalized cusps arise and stay totally geodesic under deformation. A complete description of the $\mathrm{SL_4\mathbb{R}}$-character variety $X(\Gamma,\mathrm{SL_4\mathbb{R}})$ is given: it has a unique positive-dimensional irreducible component $X_0$ defined by $\tau^2-(rs-3)\tau+r^3+s^3-6rs+9=0$, decomposing into two parts $S_1$ (orbifold) and $S_2$ (smooth); $S_1$ captures the $\mathrm{SL_3\mathbb{R}}$-type holonomies and the hyperbolic cusp, while $S_2$ contains isolated non-faithful representations. Consequently, every path of projective deformations of such an orbifold remains properly convex with the described end behavior, providing a global converse to prior results on generalized cusps under deformation. The work thus links local cusp deformations to a global deformation theory for turnover-ended hyperbolic orbifolds via a detailed analysis of $X(\Gamma,\mathrm{SL_4\mathbb{R}})$ and its slices.
Abstract
We study projective deformations of (topologically finite) hyperbolic 3-orbifolds whose ends have turnover cross section. These deformations are examples of projective cusp openings, meaning that hyperbolic cusps are deformed in the projective setting such that they become totally geodesic generalized cusps with diagonal holonomy. We find that this kind of structure is the only one that can arise when deforming hyperbolic turnover cusps, and that turnover funnels remain totally geodesic. Therefore, we argue that, under no infinitesimal rigidity assumptions, the deformed projective 3-orbifold remains properly convex. Additionally, we give a complete description of the character variety of the turnover $S^2(3,3,3)$ in SL(4,R).