Fundamental domains in the Einstein Universe

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. 1 arXiv:1307.6531v2 [math.DG]

Margulis spacetimes via the arc complex

“…Note that Drumm's strategy from Sect. 7.3 has recently been carried out in the Einstein setting in [10], although no complete disjointness criterion is known for the moment.…”

Schottky groups and maximal representations

“…A somewhat more geometric approach to generalizations of Fuchsian Schottky groups is to start out with a collection of disjoint subsets of a space X and then find a set of generators pairing these subsets according to some prescribed combinatorics. Following this idea, Schottky groups were generalized to projective linear groups acting on CP n ( [Nor86], [SV03]), affine groups acting on R 3 [Dru92], and the conformal Lorentzian group SO(3, 2) acting on the Einstein Universe [CFLD14]. We introduce a new generalization which is based on the notion of cyclic orders and follows the latter approach.…”

Crooked surfaces and anti-de Sitter geometry

“…where p 0 , p ∞ are non-incident and p i is incident to both p 0 , p ∞ for i = 1, 2. (Compare [2,12,16,4]. )…”

“…(See [2] for an expanded treatment of Einstein geometry and generalized crooked planes, which we renamed crooked surfaces.) Recently Virginie Charette, Dominik Francoeur, and Rosemonde Lareau-Dussault [4] have used crooked surfaces to build fundamental domains for Kleinian groups of conformal transformations of Ein 3 . More recently, Danciger-Guéritaud-Kassel have discussed which complete antide Sitter 3-manifolds admit fundamental domains bounded by crooked planes.…”

“…The purpose of this note is to interpret the AdS-crooked planes of [6] in terms of crooked surfaces in Ein 3 . In particular the crooked fundamental polyhedra for AdS-manifolds of [6] will extend to fundamental polyhedra in Ein 3 for Lorentzian Kleinian groups as in [4].…”