A classification of radial or totally geodesic ends of real projective orbifolds I: a survey of results

TL;DR

This survey classifies radial and totally geodesic ends of real projective orbifolds, showing that under certain conditions, only lens type or horospherical ends occur, advancing understanding of their geometric structures.

Contribution

It provides a comprehensive classification of ends of real projective orbifolds, identifying conditions that restrict ends to lens type or horospherical forms, and connects these to eigenvalue conditions of holonomy representations.

Findings

Only radial or totally geodesic ends of lens or horospherical type exist under certain conditions.

Eigenvalue conditions on holonomy representations characterize the types of ends.

The results apply to strongly irreducible properly convex real projective orbifolds.

Abstract

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. Hyperbolic manifolds with cusps and hyper-ideal ends are examples. For this, we will study the natural conditions on eigenvalues of holonomy representations of ends when these ends are manageably understandable. We will show that only the radial or totally geodesic ends of lens type or horospherical ends exist for strongly irreducible properly convex real projective orbifolds under the suitable conditions. The purpose of this article is to announce these results.