Configuration spaces of points on the circle and hyperbolic Dehn   fillings
  A purely combinatorial compactification of the configuration space of n (>4) distinct points with equal weights in the real projective line was introduced by M. Yoshida. We geometrize it so that it will be a real hyperbolic cone-manifold of finite volume with dimension n-3. Then, we vary weights for points. The geometrization still makes sense and yields a deformation. The effectivity of deformations arisen in this manner will be locally described in the existing deformation theory of hyperbolic structures when n-3 = 2, 3. 
Sadayoshi Kojima (TIT), Haruko Nishi (Kyushu) and Yasushi Yamashita   (Nara)