  Anosov representations were introduced by F. Labourie [18] for fundamental
groups of closed negatively curved surfaces, and generalized by O. Guichard and
A. Wienhard [19] to representations of arbitrary Gromov hyperbolic groups into
real semisimple Lie groups. In this paper, we focus on Anosov representations
into the identity component O0(2, n) of O(2, n) for n $\ge$ 2. Our main result
is that any Anosov representation with negative limit set as defined in [8] is
the holonomy group of a spatially compact, globally hyperbolic maximal (abbrev.
CGHM) conformally flat spacetime. The proof of the spatial compactness needs a
particular care. The key idea is to notice that for any spacetime M , the space
of lightlike geodesics of M is homeomorphic to the unit tangent bundle of a
Cauchy hypersurface of M. For this purpose, we introduce the space of causal
geodesics containing timelike and lightlike geodesics of anti-de Sitter space
and lightlike geodesics of its conformal boundary: the Einstein spacetime. The
spatial compactness is a consequence of the following theorem : Any Anosov
representation acts properly discontinuously by isometries on the set of causal
geodesics avoiding the limit set; besides, this action is cocompact. It is
stated in a general setting by O. Guichard, F. Kassel and A. Wienhard in [11].
We can see this last result as a Lorentzian analogue of the action of convex
cocompact Kleinian group on the complementary of the limit set in H n. Lastly,
we show that the conformally flat spacetime in our main result is the union of
two conformal copies of a strongly causal AdS-spacetime with boundary which
contains-when the limit set is not a topological (n -- 1)-sphere-a globally
hyperbolic region having the properties of a black hole as defined in [2], [3],
[4].
