Deformations of Margulis space-times with parabolics

TL;DR

This paper studies how Margulis space-times with parabolic elements can be deformed while maintaining proper action, and shows they can be decomposed into cells using crooked planes, supporting a conjecture in the field.

Contribution

It demonstrates that small deformations of Margulis space-times with parabolics remain proper and introduces a cell decomposition using crooked planes, partially confirming a conjecture.

Findings

Small deformations preserve properness of the action.

Existence of a disjoint, embedded crooked plane decomposition.

Partial affirmation of the Charette-Drumm-Goldman conjecture.

Abstract

Let $E$ be a flat Lorentzian space of signature $(2, 1)$. A Margulis space-time is a noncompact complete Lorentz flat $3$-manifold $E/\Gamma$ with a free isometry group $\Gamma$ of rank $g \geq 2$. We consider the case when $\Gamma$ contains a parabolic element. We show that sufficiently small deformations of $\Gamma$ still act properly on $E$. We use our previous work showing that $E/\Gamma$ can be compactified relative to a union of solid tori and some old idea of Carri\`ere in his famous work. We will show that the there is also a decomposition of $E/\Gamma$ by crooked planes that are disjoint and embedded in a generalized sense. These can be perturbed so that $E/\Gamma$ decomposes into cells. This partially affirms the conjecture of Charette-Drumm-Goldman.