Fundamental domains in the Einstein Universe

TL;DR

This paper explores the construction of fundamental domains in the 3-dimensional Einstein Universe using crooked surfaces, enabling explicit examples of properly acting groups in this conformal setting.

Contribution

It introduces the concept of crooked surfaces as boundaries of fundamental domains and demonstrates their disjointness, facilitating group actions in the Einstein Universe.

Findings

Existence of pairwise disjoint crooked surfaces in the Einstein Universe

Construction of explicit groups acting properly on open subsets

Application to Margulis spacetimes

Abstract

We will discuss fundamental domains for actions of discrete groups on the 3-dimensional Einstein Universe. These will be bounded by crooked surfaces, which are conformal compactifications of surfaces that arise in the construction of Margulis spacetimes. We will show that there exist pairwise disjoint crooked surfaces in the 3-dimensional Einstein Universe. As an application, we can construct explicit examples of groups acting properly on an open subset of that space.