A classification of radial and totally geodesic ends of properly convex real projective orbifolds

Abstract

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $SL(n+1, \mathbb{R})$ or $PGL(n+1, \mathbb{R})$. 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. The main techniques are the theory of Fried and Goldman on affine manifolds, a generalization of the work of Goldman, Labourie, and Margulis on flat Lorentzian $3$-manifolds and the work on Riemannian foliations by Molino, Carri\`ere, and so on. 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.