Topological tameness of Margulis spacetimes

Abstract

We show that Margulis spacetimes without parabolic holonomy are topologically tame. A Margulis spacetime is the quotient of the $3$-dimensional Minkowski space by a free proper isometric action of the free group of rank $\geq 2$. We will use our particular point of view that the Margulis spacetime is a manifold-with-boundary with an $\mathbb{R} P^3$-structure in an essential way. The basic tools are a bordification by a closed $\mathbb{R} P^2$-manifold with free holonomy group, and the work of Goldman, Labourie, and Margulis on geodesics in the Margulis spacetimes and $3$-manifold topology.