The classification of ends of properly convex real projective orbifolds II: Properly convex radial ends and totally geodesic ends

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 properly convex ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. For this, we will study the natural conditions on eigenvalues of holonomy representations of ends when these ends are manageably understandable. In this paper, we only study the properly convex ends. The main techniques are the Vinberg duality and a generalization of the work of Goldman, Labourie, and Margulis on flat Lorentzian $3$-manifolds. Finally, we show that a noncompact strongly tame properly convex real projective orbifold with generalized lens-type or horospherical ends satisfying some topological conditions always has a strongly irreducible holonomy group.

89 pages, 5 figures, This is the second part of the paper "A classification of radial and totally geodesic ends of properly convex real projective orbifolds", arXiv:1304.1605, which the author is dividing into three parts

References in corpus (2)

• Limits Under Conjugacy of the Diagonal Subgroup in SL(n,R)
• A convexity theorem for real projective structures

Cited by in corpus (4)

• Convex RP^2 Structures and Cubic Differentials under Neck Separation
• The space of properly-convex structures
• A classification of radial and totally geodesic ends of properly convex real projective orbifolds III: the convex but nonproperly convex and non-complete-affine radial ends
• On properly convex real-projective manifolds with Generalized Cusp