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

Suhyoung Choi|arXiv (Cornell University)|Jan 2, 2015

Geometric and Algebraic Topology35 references6 citations

TL;DR

This paper classifies properly convex radial and totally geodesic ends in real projective $n$-dimensional orbifolds by analyzing eigenvalue conditions of holonomy representations using Vinberg duality and generalizations of Goldman-Labourie-Margulis theory. It proves that noncompact, strongly tame, properly convex orbifolds with lens-type or horospherical ends under topological constraints have strongly irreducible holonomy groups.

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.

Motivation & Objective

• To classify properly convex radial and totally geodesic ends in $n$-dimensional real projective orbifolds.
• To understand the holonomy representations of such ends through eigenvalue conditions on the associated linear actions.
• To establish conditions under which the holonomy group of a noncompact, strongly tame, properly convex orbifold is strongly irreducible.
• To apply duality theory and affine actions to characterize lens-shaped and T-end structures in the projective setting.
• To extend Koszul’s openness result to lens-type end neighborhoods and analyze convex hulls and limit sets.

Proposed method

• Utilizes Vinberg duality to relate radial and totally geodesic ends via duality maps on convex cones.
• Applies generalized Anosov flow theory and asymptotically nice affine actions dual to tubular actions on end neighborhoods.
• Employs eigenvalue estimation techniques under uniform middle-eigenvalue conditions to characterize lens-shaped representations.
• Uses the developing map and holonomy homomorphism $h: ilde{rak{O}} o \mathbb{RP}^n$ to lift orbifold structures to convex domains in $\mathbb{S}^n$.
• Applies Koszul’s openness theorem to show that lens properties are open in the representation space of the fundamental group.
• Constructs convex hulls of ends and analyzes neighborhood expansions/shrinkage using affine connections and cone structures.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient eigenvalue conditions on holonomy representations for a properly convex end to be radial or totally geodesic?
• RQ2 How do Vinberg duality and dual affine actions characterize the structure of lens-shaped and T-end neighborhoods?
• RQ3 Under what topological and geometric conditions is the holonomy group of a noncompact, strongly tame, properly convex real projective orbifold strongly irreducible?
• RQ4 How do lens-type and horospherical ends behave under small deformations of the holonomy representation?
• RQ5 What is the role of the convex hull and limit set in determining the global structure of such orbifolds?

Key findings

• A noncompact, strongly tame, properly convex real projective $n$-orbifold with generalized lens-type or horospherical ends and suitable topological constraints has a strongly irreducible holonomy group.
• Lens-shaped end neighborhoods are characterized by uniform middle-eigenvalue conditions on the holonomy representation, ensuring convexity and proper discontinuity.
• The duality map exchanges properly convex radial ends (R-ends) with totally geodesic ends (T-ends), preserving key geometric and dynamical properties.
• The openness of lens properties in the representation space is established via a generalization of Koszul’s theorem, ensuring stability under small deformations.
• End neighborhoods can be expanded or shrunk while preserving convexity and properness, with the convex hull of an end being properly convex under the given conditions.
• The existence of a circle action $S_t$ on the end neighborhood, commuting with the holonomy, allows for $C^r$-small deformations of the affine structure preserving strict convexity and hessian positivity.