Projective deformations of hyperbolic 3-orbifolds with turnover ends

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 $$X(\pi _1S^2(3,3,3),\mathrm {SL_4\mathbb {R}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:mi>R</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .