150 research outputs found
Quasi-Fuchsian AdS representations are Anosov
In a recent paper, Q. M\'erigot proved that representations in SO(2,n) of
uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are
quasi-Fuchsian, i.e. are faithfull, discrete, and preserve an acausal subset in
the boundary of anti-de Sitter space. In the present paper, we prove the
reverse implication. It also includes: -- A construction of Dirichlet domains
in the context of anti-de Sitter geometry, -- A proof that spatially compact
globally hyperbolic anti-de Sitter spacetimes with acausal limit set admit
locally CAT(-1) Cauchy hypersurfaces
Pseudo-Anosov flows in toroidal manifolds
We first prove rigidity results for pseudo-Anosov flows in prototypes of
toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered
manifold is up to finite covers topologically equivalent to a geodesic flow and
we show that a pseudo-Anosov flow in a solv manifold is topologically
equivalent to a suspension Anosov flow. Then we study the interaction of a
general pseudo-Anosov flow with possible Seifert fibered pieces in the torus
decomposition: if the fiber is associated with a periodic orbit of the flow, we
show that there is a standard and very simple form for the flow in the piece
using Birkhoff annuli. This form is strongly connected with the topology of the
Seifert piece. We also construct a large new class of examples in many graph
manifolds, which is extremely general and flexible. We construct other new
classes of examples, some of which are generalized pseudo-Anosov flows which
have one prong singularities and which show that the above results in Seifert
fibered and solvable manifolds do not apply to one prong pseudo-Anosov flows.
Finally we also analyse immersed and embedded incompressible tori in optimal
position with respect to a pseudo-Anosov flow.Comment: 44 pages, 4 figures. Version 2. New section 9: questions and
comments. Overall revision, some simplified proofs, more explanation
Causal properties of AdS-isometry groups I: Causal actions and limit sets
International audienceWe study the causality relation in the 3-dimensional anti-de Sitter space AdS and its conformal boundary Ein. To any closed achronal subset in \mbox{Ein}_2 we associate the invisible domain from in AdS. We show that if is a torsion-free discrete group of isometries of AdS preserving and is non-elementary (for example, not abelian) then the action of on is free, properly discontinuous and strongly causal. If is a topological circle then the quotient space is a maximal globally hyperbolic AdS-spacetime admitting a Cauchy surface such that the induced metric on is complete. In a forthcoming paper we study the case where 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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