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

We study the causality relation in the 3-dimensional anti-de Sitter space AdS and its conformal boundary Ein_2. To any closed achronal subset $\Lambda$ in Ein_2 we associate the invisible domain $E(\Lambda)$ from $\Lambda$ in AdS. We show that if $\Gamma$ is a torsion-free discrete group of isometries of AdS preserving $\Lambda$ and is non-elementary (for example, not abelian) then the action of $\Gamma$ on $E(\Lambda)$ is free, properly discontinuous and strongly causal. If $\Lambda$ is a topological circle then the quotient space $M_\Lambda(\Gamma) = \Gamma \setminus E(\Lambda)$ is a maximal globally hyperbolic AdS-spacetime admitting a Cauchy surface $S$ such that the induced metric on f$S$ is complete. In a forthcoming paper we study the case where $\Gamma$ 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.

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Published 1 January 2008