The deformation space of real projective structures on the $(^*n_1n_2n_3n_4)$-orbifold

Abstract

For positive integers $n_i \geq 2, i = 1, 2, 3, 4$, such that $\Sigma \frac{n_i}{1} < 2$, there exists a quadrilateral $P = P_1 P_2 P_3 P_4$ in the hyperbolic plane $H^2$ with the interior angle $\frac{n_i}{\pi}$ at $P_i$. Let $\Gamma \subset Isom(H^2)$ be the (discrete) group generated by reflections in each side of $P$. Then the quotient space $H^2/\gamma$ is a differentiable orbifold of type $(^* n_1 n_2 n_3 n_4)$. It will be shown that the deformation space of $Rp^2$-structures on this orbifold can be mapped continuously and bijectively onto the cell of dimension 4 - \left$\mid$ {i$\mid$n_i = 2} \right$\mid$$.

Keywords

• deformation space;
• real projective structure;
• $^*n_1n_2n_3n_4$-orbifold

File

Download PDF

References

1. Proc. Amer. Math. Soc. v.118 Convex real projective structures on closed surfaces are closed S. Choi;W. Goldman
2. Senior Thesos, Princeton Univ. Affine manifolds and projective geometry on surfaces W. Goldman
3. Comtemp. Math.;Amer. Math. Soc.;Providence, R.I. v.74 Geomatric structures and varieties of representations, in "The Geometry of Group Representations" W. Goldman;W. Goldman(ed.); A. Magid(ed.)
4. Graduate Texts in Math. v.33 Differential Topology M. Hirsch
5. Graduate Texts in Math. v.149 Foundations of Hyperbolic Manofolds J. Ratcliffe
6. Bull. London Math. Soc. v.15 The geometries of 3-manifolds P. Scott
7. lecture notes The Geometry and Topology of 3-manifolds W. Thurston