# Margulis spacetimes via the arc complex

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## Abstract

We study *strip deformations* of convex cocompact hyperbolic surfaces, defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface. We prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation, in an essentially unique way. The infinitesimal version of this result gives a parameterization, by the arc complex, of the moduli space of Margulis spacetimes with fixed convex cocompact linear holonomy. As an application, we provide a new proof of the tameness of such Margulis spacetimes *M* by establishing the Crooked Plane Conjecture, which states that *M* admits a fundamental domain bounded by piecewise linear surfaces called crooked planes. The noninfinitesimal version gives an analogous theory for noncompact complete anti-de Sitter 3-manifolds.

## Notes

### Acknowledgments

We would like to thank Thierry Barbot, Virginie Charette, Todd Drumm, and Bill Goldman for interesting discussions related to this work, as well as François Labourie and Yair Minsky for igniting remarks. We are grateful to the Institut Henri Poincaré in Paris and to the Centre de Recherches Mathématiques in Montreal for giving us the opportunity to work together in stimulating environments.

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