[Paper Review] Fundamental polyhedra in the Einstein Universe

Virginie Charette, Dominik Francoeur+1 more|arXiv (Cornell University)|Jul 24, 2013

Geometric and Algebraic Topology7 references5 citations

TL;DR

This paper constructs fundamental polyhedra in the 3-dimensional Einstein Universe using pairwise disjoint crooked surfaces—conformal compactifications of Margulis spacetime surfaces—thereby enabling explicit constructions of discrete groups acting properly on open subsets of the space. The key contribution is proving the existence of such disjoint surfaces, which provides a geometric framework for studying proper group actions in this Lorentzian setting.

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.

Motivation & Objective

• To investigate discrete group actions on the 3-dimensional Einstein Universe using geometric fundamental domains.
• To understand the role of crooked surfaces as boundary components of these domains.
• To establish the existence of pairwise disjoint crooked surfaces in the Einstein Universe.
• To apply this geometric structure to construct explicit examples of groups acting properly on open subsets of the space.

Proposed method

• Utilizes crooked surfaces as conformal compactifications of surfaces from Margulis spacetime constructions.
• Analyzes the geometric and conformal properties of these surfaces within the 3-dimensional Einstein Universe.
• Constructs fundamental polyhedra bounded by these crooked surfaces to define group action domains.
• Employs conformal geometry and Lorentzian structure to ensure disjointness and properness of group actions.
• Applies techniques from discrete group theory and Lorentzian geometry to verify proper discontinuity of the actions.

Experimental results

Research questions

• RQ1 Can pairwise disjoint crooked surfaces be constructed in the 3-dimensional Einstein Universe?
• RQ2 How do crooked surfaces serve as boundaries for fundamental domains of discrete group actions?
• RQ3 What geometric conditions ensure that a group acts properly on an open subset of the Einstein Universe?
• RQ4 In what way do conformal compactifications of Margulis spacetime surfaces contribute to constructing such domains?

Key findings

• Pairwise disjoint crooked surfaces exist in the 3-dimensional Einstein Universe, confirming a key geometric prerequisite.
• These surfaces can be used to bound fundamental polyhedra, forming the basis for group action domains.
• The construction enables explicit examples of discrete groups acting properly on open subsets of the Einstein Universe.
• The method relies on conformal compactification techniques applied to Margulis spacetime surfaces.