## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 12, Number 1 (2008), 1-66.

### 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 $\mathrm{Ein}_2$. To any closed achronal subset $\Lambda$ in $\mathrm{Ein}_2$ 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 $S$ 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.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 12, Number 1 (2008), 1-66.

**Dates**

First available in Project Euclid: 10 January 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1199994329

**Mathematical Reviews number (MathSciNet)**

MR2369412

**Zentralblatt MATH identifier**

1151.83311

#### Citation

Barbot, Thierry. Causal properties of AdS-isometry groups I: causal actions and limit sets. Adv. Theor. Math. Phys. 12 (2008), no. 1, 1--66. https://projecteuclid.org/euclid.atmp/1199994329

#### See also

- See: Thierry Barbot. Causal properties of AdS-isometry groups II: BTZ multi-black-holes. Adv. Theor. Math. Phys., vol. 12, no. 6 (2008), 1209-1257.Project Euclid: euclid.atmp/1221834532