Causal properties of AdS-isometry groups I: Causal actions and limit sets

Abstract

We study the causality relation in the 3-dimensional anti-de Sitter space AdS and its conformal boundary Ein. To any closed achronal subset $Λ$ in ${Ein}\_2$ we associate the invisible domain $E(Λ)$ from $Λ$ in AdS. We show that if $Γ$ is a torsion-free discrete group of isometries of AdS preserving $Λ$ and is non-elementary (for example, not abelian) then the action of $Γ$ on $E(Λ)$ is free, properly discontinuous and strongly causal. If $Λ$ is a topological circle then the quotient space $M\_Λ(Γ) = Γ\backslash{E}(Λ)$ is a maximal globally hyperbolic AdS-spacetime admitting a Cauchy surface $S$ such that the induced metric on $S$ is complete. In a forthcoming paper we study the case where $Γ$ is elementary and use the results of the present paper to define a large family of AdS-spacetimes including all the previously known examples of BTZ multi-black holes.