Advanced Studies in Pure Mathematics

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

Suhyoung Choi

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Abstract

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathsf{SL}(n+1, \mathbb{R})$ or $\mathsf{PGL}(n+1, \mathbb{R})$. 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 for $n \geq 2$. 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 shape or horospherical ends exist for strongly irreducible properly convex real projective orbifolds under some suitable conditions. The purpose of this article is to announce these results.

Article information

Source
Hyperbolic Geometry and Geometric Group Theory, K. Fujiwara, S. Kojima and K. Ohshika, eds. (Tokyo: Mathematical Society of Japan, 2017), 69-134

Dates
Received: 2 January 2015
Revised: 26 November 2015
First available in Project Euclid: 4 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1538671942

Digital Object Identifier
doi:10.2969/aspm/07310069

Mathematical Reviews number (MathSciNet)
MR3728495

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 53A20: Projective differential geometry 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords
geometric structures real projective structures $\mathsf{SL}(n, \mathbb{R})$ representation of groups

Citation

Choi, Suhyoung. A classification of radial or totally geodesic ends of real projective orbifolds I: a survey of results. Hyperbolic Geometry and Geometric Group Theory, 69--134, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07310069. https://projecteuclid.org/euclid.aspm/1538671942


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