The conformal boundary of Margulis space--times.
type: Item
instance of: scholarly article
title: The conformal boundary of Margulis space--times.
zbMATH Open document ID: 1040.53078
DOI: 10.1016/S1631-073X(03)00170-5
author: Charles Frances
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-09-15T00:00:00Z
review text: Let \(\mathbb R^{1,2}/\Gamma\) be a Margulis space-time where \(\Gamma\) is an affine Schottky group. The group \(\Gamma\) acts on the Einstein universe \textbf{Ein}\(_3\) as a discrete group of conformal transformations. This action is properly discontinuous on a subset conformally equivalent to \(\mathbb R^{1,2}\), and a natural question is to know if the action remains properly discontinuous on a greater open set, intersecting the boundary \(\mathbb C_\infty\). The author answers this question in an affirmative way in the case where \(\Gamma\) is Drumm's affine Schottky group. The main result of the paper is stated as follows: For each affine Schottky group \(\Gamma\subset\text{Is}(\mathbb R^{1,2})\), the Margulis space-time \(M_\Gamma=\mathbb R^{1,2}/\Gamma\) admits a conformal boundary \(\partial M_\Gamma\). More precisely, there is a closed subset \(\Lambda_\Gamma^\infty\) of \(\mathbb C_\infty\), such that: (i) The action of \(\Gamma\) is free and properly discontinuous on \(\Omega_\Gamma=\)\textbf{Ein}\(_3\setminus\Lambda_\Gamma^\infty\), and \(\Omega_\Gamma\) is the biggest open subset of \textbf{Ein}\(_3\) containing \(\mathbb R^{1,2}\), having this property. (ii) The quotient space \((\Omega_\Gamma^\infty\cup\mathbb R^{1,2})/\Gamma\) (where \(\Omega_\Gamma^\infty =\mathbb C_\infty\backslash\Lambda_\Gamma^\infty)\) is a smooth manifold with boundary, whose interior is conformally equivalent to the Margulis space-time \(M_\Gamma\). The boundary \(\Omega_\Gamma^\infty/\Gamma\) is a finite union of cylinders \(\mathbb R\times\mathbb S^1\), each endowed with a conformal class of degenerate metrics. (iii) There is a two points compactification of \(\Omega_\Gamma^\infty/\Gamma\) denoted by \(\partial M_\Gamma\). The space \(M_\Gamma\cup\partial M_\Gamma\) is singular in these two points. Each semi-geodesic of \(M_\Gamma\) leaving any compact subset of \(M_\Gamma\), tends to a unique point of \(\partial M_\Gamma\).
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 1981359
zbMATH Keywords: Margulis space-time
zbMATH Keywords: Drumm's affine Schottky group
zbMATH Keywords: Einstein universe
zbMATH Keywords: Drumm's crooked planes
zbMATH Keywords: stereographic projection
zbMATH Keywords: discrete group of conformal transformations
zbMATH Keywords: free and properly discontinuous action
zbMATH Keywords: conformal boundary of a Lorentzian manifold
zbMATH Keywords: lightlike geodesic
zbMATH Keywords: lightcone
zbMATH Keywords: ping-pong dynamics
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
cites work: Fundamental polyhedra for Margulis space-times
cites work: Linear holonomy of Margulis space-times
cites work: COMPLETE FLAT LORENTZ 3-MANIFOLDS WITH FREE FUNDAMENTAL GROUP
cites work: The geometry of crooked planes
cites work: On fundamental groups of complete affinely flat manifolds
description: scientific article; zbMATH DE number 1981359