Margulis spacetimes via the arc complex
type: Item
instance of: scholarly article
title: Margulis spacetimes via the arc complex
zbMATH Open document ID: 1344.30035
author: Jeffrey Danciger
author: Fanny Kassel
author: François Guéritaud
published in: Inventiones Mathematicae
publication date: 2016-05-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1407.5422
review text: The authors study \textit{strip deformations} of convex cocompact hyperbolic surfaces. Such a deformation is the inverse of an operation introduced by Thurston in an unpublished manuscript [A spine for Teichmüller space, 3 pages (1986)]. (Note that the authors of the paper under review give a wrong reference). Thurston's construction shows that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple closed geodesics are shorter. This deformation was studied by \textit{A. Papadopoulos} and \textit{G. Théret} [Proc. Am. Math. Soc. 138, No. 5, 1775--1784 (2010; Zbl 1197.32006)], who called it a \textit{peeling} operation, and showed that the construction can be made uniform, by a constant which is strictly less than one. \textit{H. Parlier} had obtained a weaker one, without the uniform constant [Ann. Acad. Sci. Fenn., Math. 30, No. 2, 227--236 (2005; Zbl 1088.30039)]. In the paper under review, the authors show that any deformation of the surface that uniformly lengthens all closed geodesics can be obtained as a strip deformation in an essential way. The infinitesimal version of this result gives a parametrization by the curve complex of the moduli space of Margulis spacetimes with fixed cocompact linear holonomy. Here, a \textit{Margulis spacetime} is a quotient of the three-dimensional Minkowski space \(\mathbb{R}^{2,1}\) by an isometry group acting properly discontinuously. A Margulis spacetime is determined by a noncompact hyperbolic surface \(S\) and an infinitesimal deformation of \(S\), called a \textit{proper deformation}. The set of such proper deformations forms a symmetric cone in the Teichmüller space of \(S\). This cone was studied in [\textit{W. M. Goldman} et al., Ann. Math. (2) 170, No. 3, 1051--1083 (2009; Zbl 1193.57001)]. As an application, the authors of the paper under review provide a new proof of the tameness of Margulis spaceti…
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57S30
zbMATH DE Number: 6581677
zbMATH Keywords: hyperbolic surface
zbMATH Keywords: Teichmüller space
zbMATH Keywords: deformation
zbMATH Keywords: crooked plane
zbMATH Keywords: Margulis spacetimes
zbMATH Keywords: infinitesimal deformation
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
OpenAlex ID: W1940508349
arXiv ID: 1407.5422
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DOI: 10.1007/S00222-015-0610-Z
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description: scientific article; zbMATH DE number 6581677