Anosov AdS representations are Quasi-Fuchsian
Let be a cocompact lattice in . A representation is quasi-Fuchsian if it is faithfull, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space - a particular case is the case of Fuchsian representations, ie. composition of the inclusions and . We prove that if a representation is Anosov in the sense of Labourie (cf. [Lab06]) then it is also quasi-Fuchsian. We also show that Fuchsian representations are Anosov : the fact that all quasi-Fuchsian representations are Anosov will be proved in a second part by T. Barbot. The study involves the geometry of locally anti-de Sitter spaces: quasi-Fuchsian representations are holonomy representations of globally hyperbolic spacetimes diffeomorphic to locally modeled on .
Let , denote the identity components of respectively , . Let be a cocompact torsion free lattice in . For any Lie group let denote the space of representations of into equipped with the compact-open topology.
In the case we distinguish the Fuchsian representations: they are the representations obtained by composition of an embedding (by Mostow rigidity, there is only one up to conjugacy if ) and any faithfull representation of into . Their characteristic property is to be faithfull, discrete, and to preserve a totally geodesic copy of into .
If we relax the last condition by only requiring the existence of a -invariant topological -sphere in (in the Fuchsian case, the boundary of the -invariant totally geodesic hypersurface provides such a topological sphere), we obtain the notion of quasi-Fuchsian representation. We denote by the set of quasi-Fuchsian representations. It is well-known that is a neighborhood of Fuchsian representations in the space of representations of into . One way to prove this assertion, based on the Anosov character of the geodesic flow of the hyperbolic manifold (for definitions, see § 5.1.1) goes as follows: is the projection of the geodesic flow on . For every in , let , be the extremities in of the unique geodesic tangent to . These maps define an -equivariant map where is the diagonal. To any in attach a metric on varying with continuously and in a -equivariant way - for example, take the angular metric at , ie. the pull-back of the natural metric on by the map associating to a point in the unit tangent vector at of the geodesic ray starting from and ending at . This family of metrics satisfies the following property: given in and a tangent vector to at , the norm increases exponentially with when describes a geodesic ray with final extremity . This property has the following consequence: consider the flat bundle associated to the representation . Denote by the natural fibration. The map defined above induces a section of . The flow induces a flow on that preserves the image of . Last but not least, the existence of the metrics ensures that as a -invariant closed subset of , the image of is a -hyperbolic set (cf. § 5.1.1). When we deform , the flat bundle and the flow can be continuously deformed. The structural stability of hyperbolic invariant closed subsets ensures that for small deformations we still have a section of the flat bundle , the image of which is -hyperbolic. This section lifts to an equivariant map . It is quite straightforward to observe that must be constant along the stable leaves of the geodesic flow, ie. the fibers of . Therefore, it induces a continuous map , the image of which is the a -invariant topological -sphere in .
This kind of argument has been extended in a more general framework by F. Labourie in [Lab06]: he defined, for any pair where is a Lie group acting on a manifold , the notion of -Anosov representation (or simply Anosov representation when there is no ambiguity about the pair ). For a definition, see 5.1.1. We denote by the space of -Anosov representations. By structural stability, is an open domain, and simple, general arguments ensure that Anosov representations are faithfull, with discrete image formed by loxodromic elements. As a matter of fact, and where coincide: we sketched above a proof of one implication, but observe that the reverse implication, namely that quasi-Fuchsian representations are Anosov, is less obvious (it can be obtained by adapting the arguments given in the case in T. Barbot’s sequel to this article [Bar07]).
Anosov representations have been studied in different situations, mostly in the case , ie. the case where is a surface group:
– in [Lab06], F. Labourie proved that when is the group and the frame variety, one connected component of , the quasi-Fuchsian component, coincides with a connected component of : the Hitchin component. Moreover, he proved that these quasi-Fuchsian representations are hyperconvex, ie. that they preserve some curve in the projective space with some very strong convexity properties. In [Gui], O. Guichard then proved that conversely hyperconvex representations are quasi-Fuchsian. Beware: -Anosov representations are not necessarily quasi-Fuchsian; in other words, is not connected. See [Bar05c].
– In [BILW05], the authors also used the notion of Anosov representations for the study of representations of surface groups into the symplectic group of a real symplectic vector space with maximal Toledo invariant.
The present paper is devoted to the case where is a cocompact lattice of that we deform in . Whereas in the case of quasi-Fuchsian representations into presented above the geometry of hyperbolic space played an important role, the study of deeply involves the geometry of the Lorentzian analog of , namely the anti-de Sitter space . In Lorentzian geometry, appear some phenomena, latent in the Riemannian context, related to the causality notions. Whereas in hyperbolic space pair of points are only distinguished by their mutual distance, in the anti-de Sitter space we have to distinguish three types of pair of points, according to the nature of the geometry joining the two points: this geodesic may be spacelike, lightlike or timelike - in the last two cases, the points are said causally related.
The conformal boundary of the hyperbolic space plays an important role. Similarly, anti-de Sitter space admits a conformal boundary: the Einstein universe . It is a conformal Lorentzian spacetime, also subject to a causality notion. In the following theorem, is the subset of made of non-causally related pairs, ie. pairs of points that can be joined by a spacelike geodesic in :
Any -Anosov representation is quasi-Fuchsian.
The geometric ingredient of this Theorem is the fact that quasi-Fuchsian representations are precisely holonomy representations of Lorentzian manifolds locally modelled on which are spatially compact, globally hyperbolic (in short, GHC) (§ 2.1). In this introduction, let’s simply mention that, among many others, a characterization of these spacetimes is the fact to admit a proper time function (a time function being a function with everywhere timelike gradient). It is only recently that the relevance of this notion in constant curvature spacetimes started to be perceived, a great impetus being given by the paper [Mes07] when it was circulating in the physical and the mathematical community as well in the 90’s (see also [ABB07]). The classification of GHC spacetimes of constant curvature is one of the main motivation of the present paper (the case of constant curvature and being already treated in respectively [Sca99], [Bar05b]), and of its sequel by T. Barbot ([Bar07]) where the converse of theorem 1.1 is proved.
2.1. Basic causality notions
We assume the reader acquainted to basic causality notions in Lorentzian manifolds like causal or timelike curves, inextendible causal curves, time orientation, future and past of subsets, time function, achronal subsets, etc… We refer to [BEE96] or [O’N83, § 14] for further details.
By spacetime we mean here an oriented and time oriented manifold. A spacetime is strongly causal if its topology admits a basis of causally convex neighborhoods, ie. neighborhoods such that any causal curve with extremities in is contained in .
Recall that a spacetime is globally hyperbolic (abbreviation GH) if it admits a Cauchy hypersurface, ie. an achronal subspace which intersects every inextendible timelike curve at exactly one point - such a subspace is automatically a topological locally Lipschitz hypersurface (see [O’N83, § 14, Lemma 29]).
A globally hyperbolic spacetime is called spatially compact (in short GHC) if its Cauchy hypersurfaces are compact. An alternative and equivalent definition of GHC spacetimes is to require the existence of a proper time function.
2.2. Anti-de Sitter space
Let be the vector space of dimension , with coordinates , endowed with the quadratic form:
We denote by the associated scalar product. For any subset of we denote the orthogonal of , ie. the set of elements in such that for every in . We also denote by the isotropic cone .
The anti-de Sitter space is endowed with the Lorentzian metric obtained by restriction of .
We will also consider the coordinates with:
Observe the analogy with the definition of hyperbolic space - moreover, every subset is a totally geodesic copy of the hyperbolic space embedded in . More generally, the totally geodesic subspaces of dimension in are connected components of the intersections of with the linear subspaces of dimension in . In particular, geodesics are intersections with -planes.
We will also often need an auxiliary Euclidean metric on . Let’s fix once for all the euclidean norm defined by:
2.3. Conformal model
The anti-de Sitter space is conformally equivalent to , where is the standard riemannian metric on , where is the standard metric (of curvature ) on the sphere and is the open upper hemisphere of .
In the -coordinates the metric is:
where is the hyperbolic norm, ie. the induced metric on . More precisely, is a sheet of the hyperboloid . The map sends this hyperboloid on , and an easy computation shows that the pull-back by this map of the standard metric on the hemisphere is . The proposition follows. ∎
Proposition 2.3 shows in particular that contains many closed causal curves. But the universal covering , conformally equivalent to , contains no periodic causal curve. It is strongly causal, but not globally hyperbolic.
2.4. Einstein universe
Einstein universe is the product endowed with the metric where is as above the standard spherical metric. The universal Einstein universe is the cyclic covering equipped with the lifted metric still denoted , but where now takes value in . According to this definition, and are Lorentzian manifolds, but it is more adequate to consider them as conformal Lorentzian manifolds. We fix a time orientation: the one for which the coordinate is a time function on .
In the sequel, we denote by the cyclic covering map. Let be a generator of the Galois group of this cyclic covering. More precisely, we select so that for any in the image is in the future of .
Even if Einstein universe is merely a conformal Lorentzian spacetime, one can define the notion of photons, ie. (non parameterized) lightlike geodesics. We can also consider the causality relation in and . In particular, we define for every in the lightcone : it is the union of photons containing . If we write as a pair in , the lightcone is the set of pairs such that where is distance function for the spherical metric .
There is only one point in at distance of : the antipodal point . Above this point, there is only one point in contained in : the antipodal point . The lightcone with the points , removed is the union of two components:
– the future cone: it is the set ,
– the past cone: it is the set .
Observe that the future cone of is the past cone of , and that the past cone of is the future cone of .
According to Proposition 2.3 (respectively ) conformally embeds in (respectively ). Hence the time orientation on selected above induces a time orientation on and . Since the boundary is an equatorial sphere, the boundary is a copy of the Einstein universe . In other words, one can attach a “Penrose boundary” to such that is conformally equivalent to , where is the closed upper hemisphere of .
The restrictions of and to are respectively a covering map over and a generator of the Galois group of the covering; we will still denote them by and .
2.5. Isometry groups
Every element of induces an isometry of , and, for , every isometry of comes from an element of . Similarly, conformal isometries of are projections of elements of acting on (still for ).
In the sequel, we will only consider isometries preserving the orientation and the time orientation, ie. elements of the neutral component (or ).
2.6. Achronal subsets
Recall that a subset of a conformal Lorentzian manifold is achronal (respectively acausal) if there is no timelike (respectively causal) curve joining two distinct points of the subset. In , every achronal subset is precisely the graph of a -Lipschitz function where is a subset of endowed with its canonical metric ). In particular, the achronal closed topological hypersurfaces in are exactly the graphs of the -Lipschitz functions : they are topological -spheres.
Similarly, achronal subsets of are graphs of -Lipschitz functions where is a subset of , and achronal topological hypersurfaces are graphs of -Lipschitz maps .
Stricto-sensu, there is no achronal subset in since closed timelike curves through a given point cover the entire . Nevertheless, we can keep track of this notion in by defining “achronal” subsets of as projections of geniune achronal subsets of . This definition is justified by the following results:
The restriction of to any achronal subset of is injective.
Since the diameter of is , the difference between the -coordinates of two elements of an achronal subset of is at most . The lemma follows immediately. ∎
Let , be two achronal subsets of admitting the same projection in . Then there is an integer such that:
2.7. The Klein model of the anti-de Sitter space
We now consider the quotient of by positive homotheties. In other words, is the double covering of the projective space . We denote by the projection of on . The projection is one-to-one in restriction to . The Klein model of the anti-de Sitter space is the projection of in , endowed with the induced Lorentzian metric.
is also the projection of the open domain of defined by the inequality . The topological boundary of in is the projection of the isotropic cone ; we will denote this boundary by . By construction, the projection defines an isometry between and . The continuous extension of this isometry is a canonical homeomorphism between and .
For every linear subspace of dimension in , we denote by the corresponding projective subspace of dimension in . The geodesics of are the connected components of the intersections of with the projective lines of . More generally, the totally geodesic subspaces of dimension in are the connected components of the intersections of with the projective subspaces of dimension of .
In the conformal model, the spacelike geodesics of ending at some point of are all orthogonal to at whereas in the Klein model spacelike geodesics ending at a given point in are not tangent one to the other. Hence the homeomorphism between and is not a diffeomorphism.
For every in , the affine domain of is the connected component of containing . Let be the connected component of containing . The boundary of in is called the affine boundary of .
Up to composition by an element of the isometry group of , we can assume that is the projection of the hyperplane in and is the projection of the region in . The map
induces a diffeomorphism between and mapping the affine domain to the region . The affine boundary corresponds to the hyperboloid . The intersections between with the totally geodesic subspaces of correspond to the intersections of the region with the affine subspaces of .
Although the real number is well-defined only for , its sign is well-defined for .
Let be an affine domain in and be its affine boundary. Let be be a point in , and be a point in . There exists a causal (resp. timelike) curve joining to in if and only if (resp. ).
2.8. The Klein model of the Einstein universe
Similarly, Einstein universe has a Klein model: it is the projection in of the isotropic cone in . The conformal Lorentzian structure can be defined in terms of the quadratic form . In particular, an immediate corollary of Lemma 2.9 is:
For , the following assertions are equivalent.
is achronal (respectively acausal)
when we see as a subset of the scalar product is non-positive (respectively negative) for every distinct .
The affine boundary defined in remark 2.8, as a domain of , is conformally isometric to the de Sitter space. Hence we also call it de Sitter domain.
2.9. Unit tangent bundle
Denote by (resp. ) the tangent bundle of unit spacelike (respectively lightlike) tangent vectors. For such a vector tangent to at , the geodesic issued from has a future and past limit in the Einstein universe. We denote by the applications which maps such a vector to its limits.
3. Regular manifolds
3.1. AdS regular domains
Let be a closed achronal subset of , and be the projection of in . We denote by the invisible domain of in , that is,
where and are the causal past and the causal future of in . We denote by the closure of in and by the projection of in (according to Corollary 2.5, only depends on , not on ).
A -dimensional AdS regular domain is a domain of the form where is the projection in of an achronal subset containing at least two points. If is a topological -sphere, then is GH-regular (this definition is motivated by theorem 4.7).
For every closed achronal set in , the invisible domain is causally convex in of : this is an immediate consequence of the definitions. It follows that AdS regular domains are strongly causal.
Let be a closed achronal subset of . Recall that is the graph of a -Lipschitz function where is a closed subset of (§ 2.6). Define two functions as follows:
where is the distance induced by on . It is easy to check that
The following lemma is a refinement of lemma 2.4:
For every (non-empty) closed achronal set , the projection of on is one-to-one.
We use the notations introduced in remark 3.3. For every , there exists a point such that . Hence, for every , we have . Hence lies in The restriction to of the projection of on is obviously one-to-one. ∎
An achronal subset of is pure lightlike if the associated subset of contains two antipodal points and such that, for the associated 1-Lipschitz map the equality holds.
If is pure lightlike, for every element of we have , implying that is empty. Conversely:
is empty if and only if is pure lightlike. More precisely, if for some point in the equality holds then is pure lightlike.
Assume for some in . Then, since is compact, the upper and lower bounds are attained: there are in such that:
We are in the equality case of the triangular inequality. It follows that belongs to a minimizing geodesic in joining to . It is possible only if , are antipodal one to the other, since if not the minimizing geodesic joining them is unique and contained in . Moreover, . The lemma follows. ∎
For every achronal topological -sphere ,
is disjoint from (ie. it is contained in );
, where denotes the closure of in .
We use the notations introduced in remark 3.3. Since is a topological -sphere, the set is the whole sphere . For every , one has . Finally, recall that (resp. ) if and only if (resp. ). The corollary follows. ∎
It follows from item (2) of Corollary 3.7 that the GH-regular domain characterizes , ie. invisible domains of different achronal -spheres are different. We call the limit set of .
3.2. AdS regular domains as subsets of
The canonical homeomorphism between and allows us to see AdS regular domains as subsets of .
Let be the projection of a closed achronal subset of which is not pure lightlike. We see and in . Then and are contained in the union of an affine domain and its affine boundary.
See [Bar05a, Lemma 8.27]. ∎
Putting together the definition of the invisible domain of a set and Lemma 2.9, one gets:
Let be the projection of a closed achronal subset of which is not pure lightlike. If we see and in the Klein model , then
A nice (and important) corollary of this Proposition is that the invisible domain associated with a set is always geodesically convex: any geodesic joining two points in is contained in .
4. Globally hyperbolic AdS spacetimes
4.1. Cosmological time functions
In any spacetime , one can define the cosmological time function as follows (see [AGH98]):
The cosmological time function of a spacetime is the function defined by
where is the set of past-oriented causal curves starting at , and is the Lorentzian length of the causal curve .
This function is in general very badly behaved. For example, in the case of Minkowski space, the cosmological time function is everywhere infinite.
A spacetime is CT-regular with cosmological time function if
has finite existence time, for every in ,
for every past-oriented inextendible causal curve , .
Theorem 4.3 ([Agh98]).
CT-regular spacetimes are globally hyperbolic.
A very nice feature of CT-regularity is that is is preserved by isometries (and thus, by Galois automorphisms):
Let be a CT-regular spacetime. Let be a discrete group of isometries of preserving the time orientation and without fixed points. Then, the action of on is properly discontinuous. Futhermore, the quotient spacetime is CT-regular. More precisely, if denote the quotient map, the cosmological times and satisfy:
Sketch of proof.
clearly preserves the cosmological time and its level sets. These level sets are metric spaces on which acts isometrically, and hence, properly discontinuously. It follows quite easily that acts properly discontinuously on the entire .
The proof of the identity is straightforward: it follows from the -invariance of and the fact that inextendible causal curves in are precisely the projections by of inextendible causal curves in . ∎
4.2. GH-regular AdS spacetimes are CT-regular
Let be a non-pure lightlike topological achronal -sphere in .
The AdS regular domain is CT-regular.
Recall that is, by definition, the projection of an achronal topological sphere , and that is the projection of the invisible domain of in . We will prove that has regular cosmological time. Since the projection of on is one-to-one (lemma 3.4), this will imply that also has regular cosmological time. We denote by the cosmological time of .
Let be a point in . On the one hand, according to corollary 3.7, is a compact subset of , and the intersection equals . On the other hand, since is in the invisible domain of , the set is disjoint from . Therefore is a compact subset of . Therefore is conformally equivalent to a compact causally convex domain in , with a bounded conformal factor since everything is compact. It follows that the lengths of the past-directed causal curves starting at contained in is bounded (in other words, is finite), and that, for every past-oriented inextendible causal curve with , one has when . This proves that has regular cosmological time. ∎
4.3. GHC AdS spacetimes are GH-regular
A GH spacetime with constant curvature is maximal (abbreviation MGH) if it admits no non-surjective embedding in another GH spacetime with constant curvature such that each Cauchy hypersurface in embeds in as a Cauchy hypersurface.
Any GH spacetime with constant curvature embeds in a MGH spacetime, and this maximal extension is unique up to isometry (see [CBG69]). Hence, the classification of GH spacetimes with constant curvature essentially reduces to the classification of MGH ones.
Every -dimensional MGHC spacetime with constant curvature is isometric to the quotient of a GH-regular domain in by a torsion-free discrete subgroup of .
This theorem was proved by Mess in his celebrated preprint [Mes07, ABB07] (Mess only deals with the case where , but his arguments also apply in higher dimension). For the reader’s convenience, we shall recall the main steps of the proof (see [Bar05a, Corollary 11.2] for more details).
Sketch of proof of Theorem 4.7.
Let be -dimensional MGHC spacetime with constant curvature . In other words, is a locally modeled on . The theory of -structures provides us with a locally isometric developing map and a holonomy representation . Pick a Cauchy hypersurface in , and a lift of in . Then is an immersed complete spacelike hypersurface in . One can prove that such a hypersurface is automatically properly embedded and corresponds to the graph of a -Lipschitz function in the conformal model . Such a function extends to a -Lipschitz function defined on the closed disc . This shows that the boundary of in