This is a review text file submitted electronically to MR. Reviewer: W.M.Goldman Reviewer number: 014521 Address:

Math Dept. University of Maryland,College Park, Maryland 20742 Author: François Salein Short title: Variétés anti-de Sitter de dimension 3 possèdant un champ de Killing non trivial Control number: 1 443 988 Primary classification: 53 C Secondary classification(s): Review text:

This note classifies all compact 3-dimensional Lorentz manifolds of constant negative curvature which possess a nontrivial Killing field. Such manifolds are locally isometric to anti-de Sitter space X, a convenient model of which is the universal covering group { \atop PSL(2, ℝ)}, with isometries given by the group G = { \atop PSL(2, ℝ)} × { \atop PSL(2, ℝ)} where one factor acts by left-multiplication and the other by right-multiplication. Kulkarni and Raymond (J. Diff. Geo. 21 (1985), 113–138) gave the first examples of compact anti-de Sitter manifolds. Their examples are quotients of anti-de Sitter space by discrete groups of the form graph(ρ), where ρ : Γ ⟶ { \atop PSL(2, ℝ)} is a homomorphism taking values in a compact subgroup of { \atop PSL(2, ℝ)}, and Γ ⊂ { \atop PSL(2, ℝ)} is a discrete group. (This condition is useful since it implies that the holonomy group lies in a subgroup H ⊂ G which acts transitively and properly on X.) Furthermore, Kulkarni and Raymond showed that every properly discontinuous action with compact quotient is equivalent to the graph of a homomorphism ρ : Γ ⟶ { \atop PSL(2, ℝ)} where Γ ⊂ { \atop PSL(2, ℝ)} is a discrete cocompact subgroup.

Subsequent further examples were given by Goldman (J. Diff. Geo. 21 (1985), 301–308) and Ghys (Ann. Sci. Ècole. Norm. Sup. 20 (1987), 251– 270) all of which possess nontrivial Killing vector fields, but the holononmy does not lie in a properly acting transitive subgroup of G. The difficulty in these examples is that in general compactness of an indefinite metric does not imply completeness.

Recently Klingler (Math. Ann. 306 (1996), 353–370) proved that compact anti-de Sitter manifolds are geodesically complete; thus any compact Lorentzian manifold of constant negative curvature is a quotient of anti-de Sitter space. It follows that whenever ρ is sufficiently close to the trivial representation, graph(ρ) acts properly and cocompactly on X. The paper under review, determines explicitly how small ρ must be for graph(ρ) to act properly, in the special case ρ takes values in an abelian subgroup of { \atop PSL(2, ℝ)}. If ρ takes values in an elliptic or parabolic one-parameter subgroup, then graph(ρ) acts properly. Thus, suppose that ρ takes values in a hyperbolic one-parameter subgroup H. For γ ∈ Γ, let t(γ) the minimum displacement of γ acting on the hyperbolic plane. Then the stable norm on H^{1}(Γ, ℝ) is defined as the the function

∥h∥ = lim_{n→∞} \frac{1}{n} inf\{t(γ_{n}) | [γ_{n}] = nh\}

where [γ] denotes the equivalence class of γ ∈ Γ in H^{1}(Γ, ℝ). Identifying the H-valued representations ρ with H^{1}(Γ, ℝ), the author proves that graph(ρ) acts properly if and only if ∥ρ∥ < 1.