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

TL;DR

This survey discusses partial results on convex real projective orbifolds with radial or totally geodesic ends, focusing on deformation spaces and the openness and closedness of convex structures.

Contribution

It establishes a homeomorphism between deformation spaces of convex real projective structures with specific ends and certain strata of representation varieties.

Findings

Homeomorphism between deformation space and representation space strata.

Conditions for openness and closedness of convex structures.

Partial results on deformation theory of orbifolds with special ends.

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 the totally geodesic boundary. 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 corresponding subset of \[ Hom(\pi_{1}(\mathcal{O}), PGL(n+1, \mathbb{R}))/PGL(n+1, \mathbb{R}).\] 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.