Margulis spacetimes via the arc complex

TL;DR

This paper characterizes deformations of convex cocompact hyperbolic surfaces as strip deformations along geodesic arcs, and uses this to parameterize Margulis spacetimes via the arc complex, proving their tameness.

Contribution

It establishes a correspondence between surface deformations and the arc complex, providing a new proof of Margulis spacetime tameness and extending to anti-de Sitter manifolds.

Findings

Any lengthening deformation is a strip deformation

Parameterization of Margulis spacetimes by the arc complex

Proof of the Crooked Plane Conjecture for Margulis spacetimes

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 complete anti-de Sitter 3-manifolds.