Configuration spaces of points on the circle and hyperbolic Dehn fillings (Q1292698)

For any integer \(n \geq 5\) let \(X(n)\) be the space of configurations of \(n\) distinct points of the real projective line \(\mathbb{R} P^1\) up to projective automorphisms. Let the group \(\text{PGL}(2,\mathbb{R})\) act on \({(\mathbb{R} P^1)}^n\) diagonally and set \(D = \{ (x_1,\dots,x_n) \in {(\mathbb{R} P^1)}^n\mid x_i = x_j\) for some \(i \neq j\}\). Then \(X(n)\) can be identified with the orbit space \(({(\mathbb{R} P^1)}^n \setminus D)/\text{PGL}(2,\mathbb{R})\). By considering the real slice of Thurston's complex hyperbolization, the authors establish a real hyperbolic polyhedral structure on each of the connected components of \(X(n)\), where the boundary of each polyhedron is coded by a degenerate configuration. Using this construction, the space \(X(n)\) can be embedded as an open dense subset in some connected real hyperbolic cone-manifold. In the cases \(n=5,6\) the authors relate their geometrization to the existing deformation theory of hyperbolic structures.