The convex real projective orbifolds with radial or totally geodesic ends: A survey of some partial results

Abstract

A real projective orbifold has a radial end if a neighborhood of the end is foliated by projective geodesics that develop into geodesics ending at a common point. It has a totally geodesic end if the end can be completed to have a totally geodesic boundary. We will prove a homeomorphism between the deformation space of convex real projective structures on an orbifold O with radial or totally geodesic ends with various conditions, and the union of open subspaces of strata of a subset Hom epsilon(pi(1)(O), PGL)/ PGL of the PGL-character variety for pi(1)(O) given by corresponding end conditions for holonomy representations. Lastly, we will talk about the openness and closedness of the properly (resp. strictly) convex real projective structures on a class of orbifold with generalized admissible ends.