Margulis spacetimes via the arc complex
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 Such a deformation is the inverse of an operation introduced by Thurston in an unpublished manuscript [A spine for Teichmüller space, 3 pages (1986)].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 spacetimes under the assumption that the associated hyperbolic surface is convex cocompact. In doing so, they establish a conjecture made by \textit{T. A. Drumm} and \textit{W. M. Goldman} [Electron. Res. Announc. Am. Math. Soc. 1, No. 1, (1995; Zbl 0849.51012)]. This conjecture, called the \textit{Crooked Plane Conjecture}, states that a Margulis spacetime admits a fundamental domain by piecewise linear surfaces (called crooked planes).
scientific article; zbMATH DE number 6581677
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