Finite‐sided deformation spaces of complete affine 3‐manifolds. (31st July 2013)

• Record Type: Journal Article
• Authors: Charette, Virginie; Drumm, Todd A.; Goldman, William M.
• Is Part Of: Journal of topology. Volume 7:Part 1(2014)
• Page Start: 225
• Page End: 246
• Publication Date: 2013-07-31
• Subjects: Topology -- Periodicals; 514.05
• DOI: 10.1112/jtopol/jtt028
• Languages: English
• ISSNs: 1753-8416

Abstract

Abstract : A Margulis spacetime is a complete affine 3‐manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S . We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic to a two‐holed cross‐surface Σ, that is, the complement of two disjoint disks in ℝP 2 . We show that every such manifold is homeomorphic to a solid handlebody of genus 2, and admits a fundamental polyhedron bounded by crooked planes. Furthermore, the deformation space is a bundle of convex four‐sided cones over the space of marked hyperbolic structures. The sides of each cone are defined by invariants of the two components of ∂ Σ and the two orientation‐reversing simple curves. The two‐holed cross‐surface, together with the three‐holed sphere, are the only topologies Σ for which the deformation space of complete affine structures is finite‐sided.