Finite-sided deformation spaces of complete affine 3-manifolds
type: Item
instance of: scholarly article
title: Finite-sided deformation spaces of complete affine 3-manifolds
zbMATH Open document ID: 1297.30070
author: Virginie Charette
author: Todd A. Drumm
author: William M. Goldman
published in: Journal of Topology
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.2862
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 6340747
zbMATH Keywords: Margulis spacetime
zbMATH Keywords: affine deformations
zbMATH Keywords: Margulis invariant
zbMATH Keywords: Fricke space
zbMATH Keywords: crooked planes
MaRDI profile type: Publication
OpenAlex ID: W2150540309
arXiv ID: 1107.2862
DOI: 10.1112/JTOPOL/JTT028
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review text: Let \(E\) denote the real affine \(3\)-dimensional space endowed with the Minkowski inner product. A \textsl{Margulis spacetime} is a manifold obtained as the quotient of \(E\) by the proper action of a discrete free group \(\Gamma\) of affine Minkowski isometries. The natural linear representation of \(\Gamma\) into the group of linear Minkowski isometries \(\mathrm{SO}(2,1)\) is faithful and discrete. Its image \(\Gamma_0\) is thus a Fuchsian group, and it follows that one can naturally associate to each Margulis spacetime a hyperbolic surface with free fundamental group.NEWLINENEWLINEThe paper is devoted to the study of Margulis spacetimes whose associated hyperbolic surfaces are topologically real projective planes with two punctures. The authors prove that in this case the Margulis spacetime is homeomorphic to the interior of handlebody of genus \(2\). Moreover, these Margulis spacetimes are ``tame'' in the sense that \(\Gamma\) admits a fundamental domain bounded by ``crooked planes''. This latter result proves the ``Crooked Plane Conjecture'' for proper affine deformations of Fuchsian groups uniformising the twice-punctured real projective plane.NEWLINENEWLINESome remarks on Margulis spacetimes whose associated hyperbolic surfaces have free fundamental group of rank \(2\) (sphere with four punctures, once-punctured torus and Klein bottle) are also provided.
reviewed by: Luisa Paoluzzi
description: scientific article; zbMATH DE number 6340747