Tameness of Margulis space-times with parabolics

Abstract

Abstract Let 𝖤 {\mathsf{E}} be a flat Lorentzian space of signature ( 2 , 1 ) {(2,1)} . A Margulis space-time is a noncompact complete Lorentz flat 3-manifold 𝖤 / Γ {\mathsf{E}/\Gamma} with a free holonomy group Γ of rank 𝗀 {\mathsf{g}} , 𝗀 ≥ 2 {\mathsf{g}\geq 2} . We consider the case when Γ contains a parabolic element. We obtain a characterization of proper Γ-actions in terms of Margulis and Charette–Drumm invariants. We show that 𝖤 / Γ {\mathsf{E}/\Gamma} is homeomorphic to the interior of a compact handlebody of genus 𝗀 {\mathsf{g}} generalizing our earlier result. Also, we obtain a bordification of the Margulis space-time with parabolics by adding a real projective surface at infinity giving us a compactification as a manifold relative to parabolic end neighborhoods. Our method is to estimate the translational parts of the affine transformation group and use some 3-manifold topology.