Tameness of Margulis space-times with parabolics

TL;DR

This paper characterizes proper actions of groups with parabolic elements on 3D Lorentzian manifolds, showing they are homeomorphic to handlebodies and providing a natural compactification with a projective surface at infinity.

Contribution

It extends the understanding of Margulis space-times with parabolics by characterizing proper actions and constructing a natural bordification using topological and geometric methods.

Findings

Proper actions characterized by invariants.

Homeomorphism to handlebody interiors.

Compactification via adding a projective surface.

Abstract

Let $\mathbf{E}$ be a flat Lorentzian space of signature $(2, 1)$. A Margulis space-time is a noncompact complete flat Lorentzian $3$-manifold $\mathbf{E}/\Gamma$ with a free holonomy group $\Gamma$ of rank $\mathbf{g}, \mathbf{g} \geq 2$. We consider the case when $\Gamma$ contains a parabolic element. We obtain a characterization of proper $\Gamma$-actions in terms of Margulis and Drumm-Charette invariants. We show that $\mathbf{E}/\Gamma$ is homeomorphic to the interior of a compact handlebody of genus $\mathbf{g}$ 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.