Convex real projective orbifolds with radial or totally geodesic ends: a survey of some partial results

Suhyoung Choi

Published 2016-01-26Version 1

A real projective orbifold is an $n$-dimensional orbifold modeled on ${\mathbf R}P^n$ with the group $PGL(n+1, {\mathbf R})$. We concentrate on an orbifold that contains a compact codimension $0$ submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed $(n-1)$-dimensional orbifolds times intervals. 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 the totally geodesic boundary. The orbifold is said to be convex if any path can be homotoped to a projective geodesic with endpoints fixed. The purpose of this paper is to announce some partial results. A real projective structure sometimes admits deformations to parameters of real projective structures. We will prove a homeomorphism between the deformation space of convex real projective structures on an orbifold $\mathcal{O}$ with radial or totally geodesic ends with various conditions with the union of open subspaces of strata of the subset \[ Hom_{\mathcal E}(\pi_{1}(\mathcal{O}),PGL(n+1, {\mathbf R}))/PGL(n+1, {\mathbf R}) \] of the character variety \[ Hom(\pi_{1}(\mathcal{O}),PGL(n+1, {\mathbf R}))/PGL(n+1, {\mathbf R}) \] given by corresponding end conditions for holonomy representations. Lastly, we will prove the openness and closedness of the properly (resp. strictly) convex real projective structures on a class of orbifold with generalized admissible ends, where we need the theory of Crampon-Marquis and Cooper, Long and Tillmann on the Margulis lemma for convex real projective manifolds. The theory here partly generalizes that of Benoist on closed convex real projective orbifolds.