[Paper Review] Lorentzian Kleinian Groups

Thierry Barbot|arXiv (Cornell University)|Sep 13, 2016

Geometric Analysis and Curvature Flows14 references3 citations

TL;DR

This survey introduces Lorentzian Kleinian groups as discrete subgroups of isometries in anti-de Sitter space (AdS), extending classical Kleinian group theory by incorporating causality. It shows that achronal subgroups—those respecting causal structure—admit well-defined limit sets and proper actions on invisible domains, generalizing hyperbolic geometry to Lorentzian settings and linking to globally hyperbolic spacetimes and Teichmüller theory in 2+1 dimensions.

ABSTRACT

Classical Kleinian groups are discrete subgroups of isometries of H n. The well-known theory of Kleinian groups starts with the definition of their associated limit set in the boundary of H n , and includes the geometric properties of the quotient hyperbolic space. This approach, naively applied, fails in the Lorentzian analogue anti-de Sitter space: discrete subgroups do not act properly discontinuously, and in many cases the set of accumulation points of orbits at the conformal boundary at infinity depends on the orbit. In this survey, we point out a way to extend this classical theory by introducing causality notions: the theory of limit sets and regularity domains extend naturally to achronal subgroups. This is closely related to the notions of globally hyper-bolic spacetimes, and we present what is known about the classification of globally hyperbolic spacetimes of constant curvature. We also review the close connection revealed by G. Mess ([79]) between globally hyperbolic spacetimes of dimension 2 + 1 and Teichm{\\"u}ller space. This link can be understood via the space of time-like geodesics of anti-de Sitter space, and this space has also an interesting role, presented here, in the recent works about proper group actions on spacetimes of constant curvature.

Motivation & Objective

• To extend classical Kleinian group theory—based on hyperbolic geometry—to the Lorentzian setting using causality as a foundational concept.
• To address the failure of standard Kleinian group properties (e.g., uniform orbit limits, proper discontinuous actions) in anti-de Sitter space due to non-compact stabilizers.
• To classify globally hyperbolic spacetimes of constant curvature, particularly in 2+1 dimensions, and relate them to Teichmüller space via G. Mess’s work.
• To analyze proper group actions on spacetimes of constant curvature, especially in the context of Margulis spacetimes and Anosov representations.
• To clarify the role of timelike geodesics and foliations in characterizing discrete groups acting properly on AdS and flat Lorentzian spaces.

Proposed method

• Introduces causality notions such as achronal subgroups, which preserve the causal structure of spacetime and allow for a consistent extension of limit set theory.
• Uses the conformal boundary of anti-de Sitter space, known as the Einstein universe (Ein^{1,n-1}), as the natural setting for defining limit sets.
• Applies the theory of globally hyperbolic spacetimes (MGHC) to classify constant curvature Lorentzian manifolds, especially in the AdS, dS, and flat cases.
• Relies on the identification of AdS^{1,2} with the space of matrices in PSL(2,R) × PSL(2,R), enabling a geometric interpretation of spacetimes via representations.
• Applies the concept of invisible domains—regions not visible from the limit set—where achronal groups act properly discontinuously.
• Utilizes Anosov representations and the Margulis invariant to characterize proper actions, especially in flat and AdS spacetimes.

Experimental results

Research questions

• RQ1 How can the classical theory of Kleinian groups be extended to Lorentzian geometry, given the failure of standard properties like uniform orbit limits?
• RQ2 What role does causality play in defining well-behaved discrete group actions in anti-de Sitter space?
• RQ3 How are globally hyperbolic spacetimes of constant curvature in 2+1 dimensions classified, and what is their connection to Teichmüller space?
• RQ4 What conditions ensure that a discrete subgroup of isometries acts properly discontinuously on anti-de Sitter or flat Lorentzian space?
• RQ5 Can the structure of proper actions on spacetimes be understood via foliations by timelike geodesics or complex structures on AdS?

Key findings

• Achronal subgroups of isometries in anti-de Sitter space admit a well-defined limit set in the conformal boundary (the Einstein universe), generalizing the hyperbolic case.
• Proper discontinuous actions in AdS are possible not on the whole space, but on the invisible domain—points not visible from the limit set—enabling a geometric extension of Kleinian group theory.
• In 2+1 dimensions, globally hyperbolic spacetimes of constant curvature are classified via Teichmüller space, via the correspondence established by G. Mess between anti-de Sitter spacetimes and pairs of hyperbolic structures.
• Margulis spacetimes (quotients of R^{1,2} by discrete groups) are diffeomorphic to handlebodies and admit fundamental domains bounded by crooked planes, with properness determined by the Margulis invariant.
• Margulis AdS spacetimes, arising from Anosov representations into PSL(2,R) × PSL(2,R), are foliated by timelike geodesics and are infinitesimal limits of higher-codimension structures.
• Unlike flat Margulis spacetimes, Margulis AdS spacetimes do not necessarily admit crooked fundamental domains, indicating a structural distinction in their geometric realization.