Lorentzian Kleinian groups

Charles Frances

4 December 2003

Abstract. In this article, we introduce some basic tools for the study of
Lorentzian Kleinian groups. These groups are discrete subgroups of the
Lorentzian Möbius group O(2, n), acting properly discontinuously on some
non empty open subset of Einstein’s universe, the Lorentzian analogue of
the conformal sphere.
Mathematics Subject Classification: 22E40, 30F40, 53A30, 53C50.

Key words: Lorentzian conformal geometry, Einstein’s universe, confor-

mal dynamics, Kleinian groups.

1 Introduction
To understand an hyperbolic manifold Hn+1 /Γ (Hn+1 denotes here the
(n+1)-hyperbolic space and Γ ⊂ O(1, n+1) is a discrete group of hyperbolic
isometries), a nice and powerful tool is the dynamical study of the conformal
action of Γ on the sphere Sn . This deep relationship between hyperbolic and
conformally flat geometry has a counterpart in Lorentzian geometry, often
quoted by physicists as AdS/CFT correspondence. Let us first recall what
is the Lorentzian analogue of the pair (Hn+1 , Sn ). The (n + 1)-dimensional
Lorentzian model space of constant curvature −1 is called anti-de Sitter
space, denoted AdSn+1 (precisely, we are speaking here of the quotient of
the simply connected model AdS ^ n+1 by the center of its isometry group, see
[O’N], [Wo]). This space, like the hyperbolic space, has a conformal bound-
ary. It is called Einstein’s universe, denoted Einn , and it can be defined,
up to a two-sheeted covering, as the product S1 × Sn−1 endowed with the
conformal class of the metric −dt2 × gS n−1 . From the conformal viewpoint,
Einstein’s universe has a lot of properties reminiscent of those of the sphere.
In particular, the group O(2, n) of isometries of AdSn+1 turns out to be also
the group of conformal transformations of Einn . The understanding of an
anti-de Sitter manifold AdSn+1 /Γ thanks to the conformal dynamics of Γ
on Einn is one of the motivations for studying Lorentzian Kleinian groups,
that we define by analogy with the classical theory, as discrete subgroups

1
of O(2, n) acting freely properly discontinuously on some non empty open
subset of Einn .
Since the works of Poincaré and Klein at the end of the nineteenth cen-
tury, the classical theory of Kleinian groups has generated a great amount
of works, and progressed very far (we refer the reader to [A], [Ka], [Ma],
[MK] for an historical account and good expositions on the subject).
Other notions of Kleinian groups also appeared in other geometric con-
texts, such as complex hyperbolic and projective geometry (see for example
[Go], [SV]).
To our knowledge, nothing systematic has been done for studying Loren-
tzian Kleinian groups, so that the aim of this article is to lay some basis for
the theory. In particular, our first task is to build and study nontrivial
examples of such groups.
The first part of the paper (sections 3 and 4) is devoted to what could be
called Lorentzian Möbius dynamics, namely the dynamical study of diver-
gent sequences of O(2, n), acting on Einn . This dynamics appears reacher
than that of classical Möbius transformations on the sphere. This is essen-
tially due to the fact that O(2, n) has rank two, and the different ways to
reach infinity in O(2, n) induce different dynamical patterns for the action
on Einn . These patterns, which are essentially three, are described in sec-
tion 3, propositions 3, 4 and 5. Let us mention here two new phenomena
(with respect to the Riemannian context) illustrating the dynamical com-
plications we are confronted with. Firstly, the Lorentzian Möbius group
O(2, n) is not a convergence group for its action on Einn (roughly speaking,
a group G acting by homeomorphisms on a manifold X is a convergence
group if any sequence (gi ) of G tending to infinity admits a subsequence
with a ”north-south” dynamics, i.e a dynamics with an attracting pole p+
and a repelling one p− . See for example [A] p.40 for a precise definition).
Secondly, a discrete subgroup Γ ⊂ O(2, n) does not always act properly on
AdSn+1 .
In spite of these differences with respect to the classical theory, it is
still possible to define the limit set of a discrete subgroup Γ ⊂ O(2, n) (see
section 4). This is a closed Γ-invariant subset ΛΓ ⊂ Einn , such that the
action on the complement ΩΓ is proper. It is moreover a union of lightlike
geodesics, so that it defines naturally a Γ-invariant closed subset Λ̂Γ of Ln ,
the space of lightlike geodesics of Einn (this space is described in section
2.5). Unfortunately, the nice properties of the limit set in the classical case
of groups of conformal transformations of the sphere, are generally no longer
satisfied in our situation. For example, the limit set that we define is not, in
general, a minimal set for the action of Γ on Einn (although Λ̂Γ is sometimes
minimal for the action of Γ on Ln , see theorem 1 below). Groups Γ ⊂ O(2, n)
acting properly on AdSn+1 are those whose behaviour is closest to that of
classical Kleinian groups. They will be called groups of the first type. For
them, we get nice properties for the limit set:

2
Theorem 1. Let Γ be a Kleinian group of the first type and ΛΓ its limit
set.
(i) The action of Γ is proper on ΩΓ ∪ AdSn+1 ⊂ Einn+1 .
(ii) ΩΓ is the unique maximal element among the open sets Ω ⊂ Einn
such that Γ acts properly on Ω ∪ AdSn+1 .
(iii) If moreover Γ is Zariski dense in O(2, n), then ΩΓ is the unique
maximal open subset of Einn on which Γ acts properly, and Λ̂Γ is a minimal
set for the action of Γ on Ln .

In section 5, we give several examples of families of Lorentzian Kleinian

groups. These basic examples being constructed, it is natural to try to
combine two of them to get other more complicated examples. It is the
aim of section 6, where we prove the following result (an analogue of the
celebrated Klein’s combination theorem):

Theorem 2. Let Γ1 and Γ2 be two cocompact Lorentzian Kleinian groups,

with fundamental domains D1 and D2 . Suppose that both D1 and D2 con-
tain a lightlike geodesic. Then one can construct from Γ1 and Γ2 another
cocompact Kleinian group, isomorphic to the free product Γ1 ∗ Γ2 .

By cocompact Kleinian group, we mean a group acting properly on some

open subset of Einn , with compact quotient.
We then use theorem 2 in section 7 to construct Lorentzian Schottky
groups. The study of such groups can be carried out quite completely. The
limit set ΛΓ and the topology of the conformally flat Lorentz manifold ob-
tained as the quotient ΩΓ /Γ of the domain of properness, are made explicit
in this case, and we get:

Theorem 3. Let Γ = hs1 , ..., sg i (g ≥ 2) be a Lorentzian Schottky group.

(i) The group Γ is of the first type.
(ii) The limit set ΛΓ is a lamination by lightlike geodesics. Topologically,
it is a product of RP 1 with a Cantor set.
(iii) The action of Γ is minimal on the set of lightlike geodesics of ΛΓ .
(iv) The quotient manifold ΩΓ /Γ is diffeomorphic to the product
(g−1)] (g−1)]
S × (S1 × Sn−1 )
1 , where (S1 × Sn−1 ) is the connected sum of
(g − 1) copies of S1 × Sn−1 .

2 Geometry of Einstein’s universe

A detailed description of the geometry of Einstein’s universe can be found
in [Fr1], [Fr2] and [CK]. Also, for the reader who are not very familiar
with Lorentzian space-times of constant curvatures, good expositions can
be found in [Wo] chapter 11, and [O’N] chapter 8. In this section, we briefly
recall (without any proof) the main properties which will be useful in this
article.

3
2.1 Projective model for Einstein’s universe
Let R2,n be the space Rn+2 , endowed with the quadratic form q 2,n (x) =
−x21 − x22 + x23 + ... + x2n+2 . The isotropic cone of q 2,n is the subset of R2,n
on which q 2,n vanishes. We call C 2,n this isotropic cone, with the origin
removed. Throughout this article, we will denote by π the projection from
R2,n minus the origin, on RP n+1 . The set π(C 2,n ) is a smooth hypersurface
Σ of RP n+1 . This hypersurface turns out to be endowed with a natural
Lorentzian conformal structure. Indeed, for any x ∈ C 2,n , the restriction
of q 2,n to the tangent space Tx C 2,n , that we call q̂x2,n , is degenerate. Its
kernel is just the kernel of the tangent map dx π. Thus, pushing q̂x2,n by
dx π, we get a well defined Lorentzian metric on Tπ(x) Σ. If π(x) = π(y) the
two Lorentzian metrics on Tπ(x) Σ obtained by pushing q̂x2,n and q̂y2,n are in
the same conformal class. Thus, the form q 2,n determines a well defined
conformal class of Lorentzian metrics on Σ. One calls Einstein’s universe
the hypersurface Σ, together with this canonical conformal structure.
The intersection of C 2,n with the euclidean sphere defined by x21 + x22 +
... + x2n+2 = 1 is a smooth hypersurface Σ̂ ⊂ R2,n . One can check that
q 2,n has lorentzian signature when restricted to Σ̂, and in fact, (Σ̂, q 2,n ) is
|Σ̂
isometric to the product (S1 × Sn−1 , −dt2 + gSn−1 ). Now Einstein’s universe
is conformally equivalent to the quotient of (S1 × Sn−1 , −dt2 + gSn−1 ) by an
involution (induced by the map x 7→ −x of R2,n ).

2.2 Conformal group

In the previous projective model for Einstein’s universe, the subgroup O(2, n) ⊂
GLn+2 (R) preserving q 2,n , acts conformally on Einn . In fact, the confor-
mal group Conf(Einn ) of Einn is exactly P O(2, n). Let us now recall the
following result, which is an extension to Einstein’s universe of a classical
theorem of Liouville in Euclidean conformal geometry (see for example [CK]
[Fr3]):

Theorem 4. Any conformal transformation between two open sets of Einn

is the restriction of a unique element of P O(2, n).

2.3 Lightlike geodesics and lightcones

It is a remarkable fact of pseudo-Riemannian geometry that all the metrics
of a given conformal class have the same lightlike geodesics (as sets but not
as parametrized curves). In the case of Einstein’s universe, the lightlike
2,n
geodesics are the projections on Einn of 2-planes P ⊂ R2,n such that q|P =
1
0. So, lightlike geodesics of Einn are copies of RP .
Given a point p in Einn , the lightcone with vertex p, denoted by C(p),
is the set of lightlike geodesics containing p. In the projective model, if

4
p = π(u), with u some isotropic vector of R2,n , then C(p) is just π(P ∩C 2,n ),
where P is the degenerate hyperplane P = u⊥ (the orthogonal is taken for
the form q 2,n ). The lightcones are not smooth submanifolds of Einn . The
only singular point of C(p) is p, and C(p)\{p} is topologically R × Sn−2 .

2.4 Homogeneous open subsets

We will deal in this paper with several interesting open subsets of Einn , all
obtained by removing to Einn the projectivization of peculiar linear sub-
spaces of R2,n . We will be very brief here, and we refer to [Wo] for a more
detailed study (especially concerning de Sitter and anti- de Sitter spaces).

- Minkowski components.
Given a point p ∈ Einn , the complement of C(p) in Einn is an homogeneous
open subset of Einn , which is conformally equivalent to Minkowski space
R1,n−1 . We say that this is the Minkowski component associated to p. In
fact, we have an explicit formula for the stereographic projection identifying
Einn \C(p) and R1,n−1 (see [CK] [Fr1]).

- De Sitter and anti-de Sitter components.

Just as Minkowski space arises by removing to Einn the projectivization of
a lightlike hyperplane, one also gets interesting open subsets by removing
the projectivization of other (i.e nondegenerate) hyperplanes.
If P is some hyperplane of R2,n , with Lorentzian signature, then π(P ∩
2,n
C ) is a Riemannian sphere S of codimension one. The canonical con-
formal structure of Einn induces on this sphere the canonical Riemannian
conformal structure. The stabilizer of S in O(2, n) is a group G isomorphic
to O(1, n). The complement of S in Einn is an homogeneous open subset of
Einn , conformally equivalent to the de Sitter space dSn . So, Sn−1 , with its
canonical conformal structure, appears as the conformal boundary of dSn
If P is some hyperplane of R2,n , with signature (2, n − 1), then the pro-
jection π(P ∩C 2,n ) is a codimension one Einstein’s universe E. The stabilizer
of E in O(2, n) is a subgroup isomorphic to O(2, n − 1). The complement of
E is an homogeneous open subset of Einn which is conformally equivalent to
the anti-de Sitter space AdSn . In this way, we see Einn−1 as the conformal
boundary of AdSn .

- Complement of a lightlike geodesic.

What do we get if we remove from Einn the projectivization of a maximal
isotropic subspace of R2,n ? Such subspaces are 2-planes, so that the result-
ing open set is the complement Ω∆ of a lightlike geodesic ∆ ⊂ Einn . Open
ses like Ω∆ admit a natural foliation by degenerate hypersurfaces, and this
foliation H∆ is preserved by the whole conformal group of Ω∆ . This foliation
can be described as follows: given a point p ∈ ∆, we consider the lightcone

5
C(p) with vertex p. Since ∆ is a lightlike geodesic, we have ∆ ⊂ C(p).
Now, the intersection of C(p) with Ω∆ is a degenerate hypersurface of Ω∆ ,
diffeomorphic to Rn−1 . We call it H(p). If p 6= p0 , H(p) and H(p0 ) only
intersect along ∆, and the leaves of the foliation H∆ are just the H(p) for
p ∈ ∆.

2.5 The space Ln of lightlike geodesics of Einn

Since this space will appear naturally when we will define the limit set of a
Lorentzian Kleinian group, we briefly describe it.
The stabilizer of a lightlike geodesic in O(2, n) is a closed parabolic sub-
group P , isomorphic to (R × SL(2, R) × O(n − 2)) n Heis(2n − 3), where
Heis(2n − 3) denotes the Heisenberg group of dimension 2n − 3. Thus, Ln
can be identified with the homogeneous space O(2, n)/P , which has dimen-
sion 2n − 3.
Let H ⊂ R2,n be an hyperplane with Lorentzian signature, and let Σ be
the projection of H ∩ C 2,n on Einn . The hypersurface Σ is a codimension
one Riemannian sphere of Einn . Now, for any isotropic 2-plane P ⊂ R2,n ,
P ∩ H is one dimensional and isotropic. Equivalently, any lightlike geodesic
of Einn intersects Σ in exactly one point. We get a well defined submersion
p : Ln → Σ. The fiber over q ∈ Σ is the set of lightlike geodesics inside
the lightcone C(q). So, Ln is topologically a Sn−2 fiber bundle over Sn−1 .
Notice that O(2, n) does not preserve the bundle structure.

3 Conformal dynamics on Einstein’s universe

3.1 Cartan decomposition of O(2, n)
From now, it will be more convenient to work in a basis of R2,n for which
q 2,n (x) = −2x1 xn+2 + 2x2 xn+1 + x23 + ... + x2n . We call O(2, n) the subgroup
of GLn+2 (R) preserving the form q 2,n
Let A+ be a the subgroup of diagonal matrices in O(2, n) of the form :
 λ 
e

 eµ 


 1 


 . .. 

 

 1 

 e −µ 
e−λ

with λ ≥ µ ≥ 0. Such an A+ is usually called a Weyl chamber. The group

SO(2, n) can be written as the product KA+ K where K is a maximal com-
pact subgroup of SO(2, n). This decomposition is known as the Cartan

6
decomposition of the group SO(2, n) (compare [B], [IW]). Such a decom-
position also exists for O(2, n), with K a compact set of O(2, n). Moreover,
for every g ∈ O(2, n), there is a unique a(g) ∈ A+ such that g ∈ Ka(g)K.
The element a(g) is called the Cartan projection of g. As a matrix, it is
written:

eλ(g)
 

 eµ(g) 


 1 

a(g) = 
 .. 
 . 


 1 

 e−µ(g) 
e−λ(g)
The reals λ(g) ≥ µ(g) ≥ 0 are called the distortions of the element g
(associated with the given Cartan decomposition).

3.2 Qualitative dynamical description

We want to understand the possible dynamics for divergent sequences (gk )
of O(2, n) (i.e sequences leaving every compact subset of O(2, n)). Our
approach considers sequences gk (xk ), where (xk ) is a converging sequence
of Einn . It is important to consider arbitrary such convergent sequences,
not only constant sequences, in order to characterize proper actions. Recall
that given a subgroup Γ of homeomorphisms of a manifold X, one says that
the action of Γ on X is proper if for all convergent sequences (xk ) of X and
all divergent sequences (gk ) of Γ, the sequence gk (xk ) does not have any
accumulation point in X. Notice that there exist actions for which gk (x)
diverges for all divergent (gk ) ∈ Γ and all x ∈ X, but which are not proper
(look, for example, at the action of an hyperbolic linear transformation of
SL(2, R) on the punctured plane R2 \{0}).

Definition 1. Let (gk ) be a divergent sequence of homeomorphisms of a

manifold X (i.e (gk ) leaves any compact subset of Homeo(X)). For any
point x ∈ X, we define the set:
[
D(gk ) (x) = { accumulation points of (gk (xk ))}
xk →x

The union is taken over all sequences convergingSto x.

Further, for any set E ⊂ X, D(gk ) (E) = x∈E D(gk ) (x). Taking the
union, over all divergent sequences (gk ) ∈ Γ, of the sets D(gk ) (E), we get a
closed set DΓ (E) ⊂ X, that we call the dynamic set of E.
Notice that for two points x and y in X, y ∈ DΓ (x) if and only if
x ∈ DΓ (y). We say in this case that x and y are dynamically related.

7
 The interest of this definition for the study of actions of discrete groups
can be illustrated by the following: let Γ be a discrete group of Homeo(X)
acting on some open subset Ω ⊂ X. Then, one proves easily:
Proposition 1. The group Γ acts properly on Ω iff no two points of Ω are
dynamically related.
Assuming that the action of Γ on Ω is proper, we also have:
Proposition 2. If the action of Γ on Ω has compact quotient, then every
x ∈ ∂Ω must be dynamically related to some point y of Ω (depending on x).
Now, let (gk ) be a divergent sequence of O(2, n). We define λk = λ(gk ),
µk = µ(gk ) and δk = λk − µk . We say that the sequence (gk ) tends simply
to infinity when:
a) the three sequences (λk ), (µk ) and (δk ) converge respectively to some
λ∞ , µ∞ and δ∞ in R.
b) compact factors in the Cartan decomposition of (gk ) both admit a
limit in K.
Of course, every sequence tending to infinity admits some subsequence
tending simply to infinity, so that we will restrict our study to these last
ones. The sequences tending simply to infinity split into three categories:
(i) Sequences with balanced distortions
This name denotes the sequences (gk ) for which λ∞ = µ∞ = +∞ and δ∞ is
finite.
(ii) Sequences with bounded distortion
This denotes the sequences (gk ) for which µ∞ 6= +∞.
(iii) Sequences with mixed distortions
This denotes the sequences (gk ) for which λ∞ = µ∞ = δ∞ = +∞.

To each type corresponds, as we will see soon, distinct dynamical behaviours.

Notations 1. In the following, we will use notations such as C(p), H∆ ...
We invite the reader to look at section 2, where these notation were intro-
duced.
For any set E in R2,n , we use the notation π̌(E) for π(E ∩ C 2,n ). If y and
 are two real numbers, we write I (y) for the closed interval [y − , y + ].
For every x = (x1 , x2 , ..., xn+2 ) in R2,n , we define the -box centered at x
as:
B (x) = I (x1 ) × I (x2 ) × ... × I (xn+2 )
For a sequence (gk ) of O(2, n) tending simply to infinity, we call B∞ (x) the
compact set obtained as the limit (for the Hausdorff topology) of the sequence
of compact sets gk ◦ π̌(B (x)) (this limit will always exist in the examples we
will deal with).
Lastly, we will often denote in the same way an element of O(2, n) and the
conformal transformation of Einn that it induces.

8
3.2.1 Dynamics with balanced distortions
Proposition 3. Let (gk ) be a sequence of O(2, n) with balanced distor-
tions. Then we can associate to (gk ) two lightlike geodesics ∆+ and ∆−
called attracting and repelling circles of (gk ), and two submersions
π+ : Einn \∆− → ∆+ (resp. π− : Einn \∆+ → ∆− ), whose fibers are
the leaves of H∆− (resp. H∆+ ), such that :
For every compact subset K of Einn \∆− (resp. Einn \∆+ ), D(gk ) (K) =
π+ (K) (resp. D(g−1 ) (K) = π− (K)).
k

Remark 1. Before begining the proof, let us remark that if (gk ) has balanced
distortions (resp. bounded distortion, resp. mixed distortions), it will be so
(1) (2) (1)
for any compact perturbation of (gk ), i.e any sequence (lk gk lk ) for (lk )
(2)
and (lk ) two converging sequences of O(2, n). In the same way, the conclu-
sions of the above proposition are not modified by a compact perturbation,
even if of course, π± and ∆± are. So in the following (and also for sections
3.2.2 3.2.3), we will restrict the proofs to the case where (gk ) is a sequence
of A+ .

Proof : We restrict the proof to the case λk = µk , so that δ∞ = 0.

We begin by defining ∆± and π ± . Let us call P + (resp. P − ) the 2- plane
spaned by e1 and e2 (resp. en+1 and en+2 ), and ∆+ (resp. ∆− ) the
projection on Einn of these 2- planes. The space R2,n splits as a direct sum
P+ ⊕ P0 ⊕ P− , where P0 is the span of e3 , ..., en . This splitting defines a
projection π̃+ (resp. π̃− ) from R2,n to the plane P + (resp. P − ). The image
π̃+ (x) is nonzero as soon as x is an isotropic vector of q 2,n which is not in P − .
Thus π̃+ induces a projection π+ of Einn \∆− on ∆+ whose fibers are the
projections on Einn of the fibers of π̃ + . These are degenerate hyperplanes
of R2,n , defined as q 2,n -orthogonals of vectors of P − . So, the fibers of π+
are the intersections of Einn \∆− with the lightcones with vertex on ∆− , i.e
the leaves of H∆− .
Now, let us choose x such that π(x) 6∈ ∆− . Since gk ◦ π̌(B (x)) =
π̌(Ieλk  (eλk x1 ) × Ieµk  (eµk x2 ) × I (x3 ) × ... × I (xn ) × Ie−µk  (e−µk xn+1 ) ×
Ie−λk  (e−λk xn+2 )), we get that for  sufficiently small, we have B∞ (x) =
π̌(I (x1 ) × I (x2 ) × {0} × ..... × {0})
We thus have B∞ (x) ⊂ ∆+ . Since  is arbitrarily close to 0, for any se-
quence (xk ) such that π(xk ) tends to π(x), we have limk→∞ gk ◦ π(xk ) =
π(x1 , x2 , 0, ..., 0). This concludes the proof.


3.2.2 Dynamics with bounded distortion

Proposition 4. Let (gk ) be a sequence of O(2, n) with bounded distortions.
Then we can associate to (gk ) two points p+ and p− of Einn called attract-
ing and repelling poles of (gk ), and a diffeomorphism ĝ∞ from the space

9
of lightlike geodesics of C − = C(p− ) in the space of lightlike geodesics of
C + = C(p+ ), conformal with respect to the natural conformal structure of
these two spaces, such that :
(i) For all compact subset K inside Einn \C − , we have D(gk ) (K) = {p+ }.
(ii) For a lightlike geodesic ∆ ⊂ C − and a point x of ∆ distinct from
p− , D(gk ) (x) is the lightlike geodesic ĝ∞ (∆).
(iii) The set D(gk ) (p− ) is the whole Einn .

The cones C + and C − are called attracting and repelling cones of

(gk ).

Remark 2. The dynamical pattern of the sequence (gk−1 ) is obtained by

switching the +’s and the −’s in the statement. This remark holds also for
proposition 5.

Proof : Following the remark 1, we do the proof for a sequence (gk ) of A+ ,

with limk→∞ λk = +∞.
We define p+ = π(e1 ), p− = π(en+2 ), C + = π̌((e1 )⊥ ), C − = π̌((en+2 )⊥ ).
Let us first remark that if x1 6= 0 then clearly B∞ (x) = p+ . It proves the
point (i), as well as the (iii), passing to the complement.
If π(x) ∈ C − , and for  sufficiently small, we get that B∞ (x) = π̌(R ×
Ieµ∞  (eµ∞ x2 ) × I (x3 ) × ..... × I (xn ) × Ie−µ∞  (e−µ∞ xn+1 ) × {0}).
The lightlike geodesics of C + and C − are parametrized by a sphere Sn−2
correponding to isotropic directions of the space spaned by e2 , ..., en+1 .
We define ĝ∞ as the element of O(1, n − 1) given by the matrix :
 µ∞ 
e

 1 

ĝ∞ = 
 .. 
 . 

 1 
e −µ ∞

The spaces of lightlike geodesics of C + and C − have a canonical conformal

Riemannian structure and we see that the map ĝ∞ is a conformal diffeomor-
phism between these two spaces.
By the above formula, if π(xk ) converges to π(x), the accumulation
points of the sequence gk (π(xk )) are in every B∞ (x), for arbitrally small
. The intersection of all B∞ (x) is π̌(R × {eµ∞ x2 } × {x3 } × ..... × {xn } ×
{e−µ∞ xn+1 } × {0}), i.e the image by ĝ of the lightlike geodesic passing
through p− and π(x). Conversely, every point π(y) of this geodesic is in the
Hausdorff limit of gk ◦ π̌(B (x)). Hence, there exists a sequence xk of B (x)
with limk→∞ gk ◦ π(xk ) = π(y). Let k be a sequence tending to 0. Then
limk→∞ gk ◦ π(xnkk ) = π(y) for some sequence of integers nk , and π(xnkk )
tends to π(x). This concludes the proof of (ii). 

10
3.2.3 Mixed dynamics
Proposition 5. Let (gk ) be a sequence of O(2, n) with mixed distortions.
Then we can associate to (gk ) two points p+ and p− called attracting and
repelling poles of the sequence, as well as two lightlike geodesics ∆+ et
∆− (called attracting and repelling circles), with the inclusions p+ ∈
∆+ ⊂ C + = C(p+ ) and p− ∈ ∆− ⊂ C − = C(p− ), such that the following
properties hold:
(i) For every compact subset K inside Einn \C − , the set D(gk ) (K) is
{p+ }.
(ii) If x is a point of C − , not on ∆− , then D(gk ) (x) is the lightlike
geodesic ∆+ .
(iii) If x is a point of ∆− , distinct from p− , then D(gk ) (x) is the attracting
cone C + .
(iv) The set D(gk ) p− is the whole Einn .

The cones C + and C − are called attracting and repelling cones of

the sequence (gk ).

Proof : Once again, we suppose that (gk ) is in A+ .

We define p+ = π(e1 ), p− = π(en+2 ), C + = π̌((e1 )⊥ ), C − = π̌((en+2 )⊥ ).
The circle ∆+ (resp. ∆− ) is the projection of the 2-plane spaned by e1 and
e2 (resp. en+1 and en+2 ). We don’t prove points (i) and (iv), the proof
being exactly the same as for proposition 4.
If π(x) ∈ C − but π(x) 6∈ ∆− , it means that x1 = 0 but x2 6= 0. In this
case, we get for B∞ (x) = π̌(R × I (x2 ) × {0} × ..... × {0}), that is to say ∆+ .
The intersection of all the B∞ (x) is π̌(R × {x2 } × {0} × ..... × {0}), i.e
the lightlike geodesic ∆+ . The fact that D(gk ) (π(x)) = ∆+ is proved exactly
as in proposition 4.
When π(x) ∈ ∆− , only xn+1 and xn+2 don’t wanish and by the asump-
tion π(x) 6= p+ , we get xn+1 6= 0. Hence, we have that B∞ (x) is π̌(R × ... ×
R × I (xn+1 ) × {0}), that is to say C + .
As previously, we get D(gk ) (π(x)) = C + .


Remark 3. Notice that different configurations for the dynamical elements

described above can occur. For example, attracting and repelling circles of
a dynamics with balanced or mixed distortions can intersect, or even be the
same. In fact, all the possible configurations can occur.

11
4 About the limit set of a Lorentzian Kleinian
group
4.1 Definition of the limit set
Given a Kleinian group Γ on a manifold X, it is quite natural to ask if there
is in some sense a “canonical” open set Ω ⊂ X on which Γ acts properly.
For example, any Kleinian group Γ on the sphere Sn admits a limit set ΛΓ
and the open set ΩΓ = Sn \ΛΓ is distinguished, since it is the only maximal
open subset on which Γ acts properly. The nice properties of the limit set
of a Kleinian group on Sn rest essentially on the fact that the Möbius group
O(1, n + 1) is a convergence group on Sn . We just saw in the previous
section that O(2, n) is quite far from being a convergence group on Einn
but nevertheless, we would like to define a limit set ΛΓ associated to a given
discrete group Γ ⊂ O(2, n). We require that such a limit set have at least
the two following properties:
(i) ΛΓ is a Γ-invariant closed subset of Einn .
(ii) The action of Γ on ΩΓ = Einn \ΛΓ is properly discontinuous.

Definition 2. Given Γ discrete in O(2, n), we define SΓ (resp. TΓ ) the

set of sequences (γk ) of Γ, tending simply to infinity, with mixed or balanced
distortions (resp. with bounded distortion). If (γk ) is a sequence of SΓ (resp.
TΓ ), we call ∆+ (γk ) and ∆− (γk ) (resp. C + (γk ) and C − (γk )) its attracting
and repelling circles (resp. attracting and repelling cones).

Definition 3. We define the limit set of a discrete Γ ⊂ O(2, n) as:

(1) (2)
ΛΓ = ΛΓ ∪ ΛΓ where

(1)
[
ΛΓ = ∆+ (γk ) ∪ ∆− (γk )
(γk )∈SΓ

and

(2)
[
ΛΓ = C + (γk ) ∪ C − (γk )
(γk )∈TΓ

Notation 1. The complement of ΛΓ in Einn is denoted by ΩΓ .

It is clear that ΛΓ is closed and Γ-invariant. Let us remark that ΛΓ is a

union of lightlike geodesics, so that it also defines a closed Γ-invariant subset
Λ̂Γ ⊂ Ln .
From the dynamical properties stated in the previous section, one checks
easily that no pair of points in ΩΓ can be dynamically related, so that the
action of Γ on ΩΓ is proper.

12
4.2 Lorentzian Kleinian groups of the first and the second
type
Until now, we didn’t focus on a fundamental difference between the action
of O(1, n + 1) on Sn and that of O(2, n) on Einn . Although any discrete
group Γ ⊂ O(1, n + 1) automatically acts properly on Hn+1 , it is not true
in general that a discrete Γ ⊂ O(2, n) does so on AdSn+1 . This motivates
the following distinction between subgroups of O(2, n):
Definition 4. A discrete group Γ of O(2, n) is of the first type if it acts
properly on AdSn+1 . If not, it is said to be of the second type.
Notice that this terminology has no connection with the denomination
first kind and second kind for the standard Kleinian groups on the sphere.
The previous dichotomy has a nice translation into dynamical terms
thanks to the:
Proposition 6. A Kleinian group Γ of O(2, n) is of the first type if and
only if it doesn’t admit any sequence (γk ) with bounded distortion.
Proof : We endow R2,n+1 with the quadratic form q 2,n+1 (x) = −2x1 xn+2 +
2x2 xn+1 + x23 + ... + x2n + x2n+3 and call e1 , ..., en+3 the canonical basis.
The subgroup of O(2, n + 1) leaving invariant the subspace spaned by the
first n + 2 basis vectors can be canonically identified with O(2, n). This
identification defines an embedding j from O(2, n) into O(2, n + 1). The
action of j(O(2, n)) on Einn+1 leaves invariant a codimension 1 Einstein
universe that we call Einn . As we saw in the introduction, the complement
of Einn in Einn+1 is conformally equivalent to the anti-de Sitter space
AdSn+1 .
 Let us  consider some g in O(2, n). In the basis e1 , ..., en+3 , j(g) =
g
, so that when we perform the Cartan decomposition of j(g), we
1
find the same distortions as for g.
Suppose now that Γ admits some sequence (γk ) with bounded distortion.
By the remark above, j(γk ) has also bounded distortion as a sequence of
O(2, n+1). We call C + and C − its attracting and repelling cones in Einn+1 .
By proposition 4, D(gk ) (C − ∩ AdSn+1 ) = C + ∩ AdSn+1 . Therefore, we can
find two points of AdSn+1 which are dynamically related, so that the action
of (γk ) on AdSn+1 can not be proper (proposition 1).
Conversely, let us consider some sequence (γk ) tending simply to infinity
and with balanced or mixed distortions. Then the sequence j(γk ) has the
same properties. Let us call ∆+ and ∆− the attracting and repelling circles
of this latter sequence. Looking at the matrix expressions, it is clear that
∆+ ⊂ Einn and ∆− ⊂ Einn . By propositions 3 and 5, D(gk ) (x) ⊂ Einn
for any point x ∈ AdSn+1 . So, if we assume that Γ has no sequence with
bounded distortion, we get DΓ (x) ⊂ Einn for any point x ∈ AdSn+1 . Using
proposition 1, we get that Γ acts properly on AdSn+1 . 

13
4.3 Limit set of a group of the first type: proof of Theorem
1
Since Γ is of the first type, ΛΓ is also the limit set of Γ, seen as a subgroup
of O(2, n + 1) acting on Einn+1 . The complement of this limit set in Einn+1
is precisely ΩΓ ∪ AdSn+1 , so that point (i) of the theorem is clear.
To prove point (ii), let us suppose that Γ acts properly on some Ω ∪
AdSn+1 with Ω not included in ΩΓ . Then there is a sequence (γk ) of Γ
(with balanced or mixed distortions) such that ∆− (γk ) meets Ω.

Lemma 1. Let Γ be a discrete group of O(2, n) acting properly on some open

set Ω. Then for any sequence (γk ) of Γ with balanced distortions, neither
∆+ (γk ) nor ∆− (γk ) meets Ω.

Proof. Suppose on the contrary that for some (γk ) with balanced distor-
tions, we have ∆+ (γk ) ∩ Ω 6= ∅. From proposition 3, we infer that the set
D(γk ) (∆+ (γk ) ∩ Ω) contains a lightlike geodesic ∆ in its interior. So, there
is a tubular neighbourhood U of ∆ contained in Ext(Ω) (Ext(Ω) denotes
the complement of Ω in Einn ). But we also infer from proposition 3 that
for any ∆ not meeting ∆− (γk ), we have limk→+∞ γk (∆) = ∆+ (γk ). As
a consequence, any lightlike geodesic of Ext(Ω) has to cut ∆− (γk ). Since
all the lightlike geodesics included in U can’t all meet ∆− (γk ), we get a
contradiction.

The lemma above tells us that the sequence (γk ) has mixed distortions.
For any point x ∈ ∆− (γk ) ∩ Ω, we have D(γk ) (x) = C + (γk ). Since C + (γk )
meets AdSn+1 , we get pairs of points in Ω ∪ AdSn+1 which are dynamically
related, and the action can’t be proper, by proposition 1.

Remark 4. For Γ Kleinian of the first type, the manifold ΩΓ /Γ appears as

the conformal boundary of the complete anti-de Sitter manifold AdSn+1 /Γ
(see [Fr4] for more details on this point).

To prove point (iii), we begin by showing that Λ̂Γ ⊂ Ln is a minimal

set. This is in fact a particular case of a general result of Benoist ([B]), but
we give a simple proof.
Let Λ̂ be a closed Γ-invariant subset of Ln . Any sequence (γk ) tending
simply to infinity in Γ has either mixed or balanced distortions. As a simple
consequence of propositions 3 and 5, we get that if ∆ is a lightlike geodesic
of Einn which does not meet ∆− (γk ), then limk→+∞ γk (∆) = ∆+ (γk ). So,
if for any sequence (γk ) as above, no geodesic of Λ̂ meets ∆− (γk ), we have
ΛΓ ⊂ Λ, and we are done.
On the contrary, if for some (γk ), all the geodesics of Λ̂ meet ∆− (γk ), we
claim that Γ can’t be Zariski dense. Indeed, by Zariski density, Γ can’t leave
∆− (γk ) invariant. So, let us choose γ ∈ Γ such that γ(∆− (γk )) 6= ∆− (γk ).
If γ(∆− (γk )) and ∆− (γk ) are disjoint, the set of lightlike geodesics meeting

14
both g(∆− (γk )) and ∆− (γk ) is contained in a two dimensional Einstein’s
universe, which have to be fixed by Γ: a contradiction with the Zariski
density of Γ.
If g(∆− (γk )) and ∆− (γk ) meet in one point p, then any lightlike geodesic
meeting both g(∆− (γk )) and ∆− (γk ) has to contain p. Indeed, due to the
fact that the quadratic form q 2,n can’t have some 3 dimensional isotropic
subspace, there is no nontrivial triangle of Einn , whose edges are pieces of
lightlike geodesics. We infer that Γ has to fix the lightcone C(p) and we get
once again a contradiction.

We can now prove that ΩΓ is the maximal open set on which the action
of Γ is proper. Suppose that Γ acts properly on Ω which is not included
in ΩΓ . We call Λ the complement of Ω in Einn . Since ΛΓ 6⊂ Λ, there is a
sequence (γk ) tending simply to infinity in Γ with ∆+ (γk ) ∩ Ω 6= ∅.

Lemma 2. If an infinite Kleinian group Γ ⊂ O(2, n) acts properly on some

open subset Ω, then the complement Λ of Ω in Einn contains a lightlike
geodesic.

Proof. Let us pick a sequence (γk ) tending simply to infinity in Γ. Suppose

first that (γk ) has mixed dynamics. Suppose that ∆− (γk ) meets Ω at a point
x (if ∆− (γk ) ∩ Ω = ∅, we are done). By properness, D(gk ) (x) ∩ Ω = ∅. But
D(gk ) (x) = C + (gk ), which contains infinitely many lightlike geodesics, and
the conclusion holds.
Also, if (gk ) has balanced (resp. bounded) distorsions, the dynamic set
D(gk ) x of x ∈ ∆− (γk ) (resp. x ∈ C − (γk )) contains infinitely many lightlike
geodesics. The proof works thus in the same way.

Now, let us look at the lightlike geodesics of Λ. Since by Zariski density,

Γ can’t fix a finite family of lightlike geodesics, there are infinitely many
lightlike geodesics in Λ. But all these geodesics have to meet ∆− (γk ), be-
cause if some ∆ does not, limk→+∞ γk (∆) = ∆+ (γk ). A contradiction with
∆+ (γk ) ∩ Ω 6= ∅. Now, we conclude as for proving the minimality property
of Λ̂Γ : all the lightlike geodesics of Λ are in the same Γ-invariant Einstein
torus, or the same Γ-invariant lightcone and we get a contradiction with the
Zariski density of Γ.

5 Some examples of Lorentzian Kleinian groups

5.1 Examples arising from structures with constant curva-
ture
In Lorentzian geometry, a completeness result ensures that any compact
Lorentzian manifold with constant sectional curvature is obtained as a quo-

15
tient R1,n−1 /Γ or AdS
] n /Γ, where Γ is a dicrete group of Lorentzian isome-
tries. This deep theorem was first proved for the case of curvature zero
by Carrière in [Ca], and generalized by Klingler in [Kl] (note that compact
Lorentzian manifolds can not have curvature +1). Another result, known
as finiteness of level (see [KR], [Ze]), ensures that any compact quotient
] n /Γ̃ (where Γ̃ is a discrete group of isometries) is in fact, up to finite
AdS
cover, a quotient AdSn /Γ. Since we saw in section 2 that R1,n−1 and AdSn
both embed conformally into Einn , and thanks to theorem 4, we get that
any compact Lorentzian structure with constant curvature is (up to finite
cover) uniformized by a Lorentzian Kleinian groups. Moreover, in this case,
the structure of the groups involved is fairly well understood, thanks to
[CaD], [Sa] and [Ze].

5.2 Examples arising from flat CR-geometry

Let us consider the complex vector space Cn+1 , endowed with the hermitian
form h1,n−1 (z) = −|z1 |2 + |z2 |2 + |z3 |2 + ... + |zn+1 |2 . We consider CC1,n , the
lightcone defined as {z ∈ Cn+1 | h1,n (z) = 0}, and call Ω− the open set
{z ∈ Cn+1 | h1,n (z) < 0}. If we project Ω− on the complex projective
space CPn , we get the complex hyperbolic space HnC . If we project CC1,n
minus the origin on CPn , we get a sphere S2n−1 , naturally endowed with
a CR-structure. This CR-sphere can be seen at the infinity of HnC . If,
instead of looking at the complex directions of CC1,n , we consider the quotient
CC1,n /R? of CC1,n by the real homotheties, then the space that we get is
Einstein’s universe of dimension 2n. In other words, there is a fibration
f : Ein2n → S2n−1 whose fibers are circles. The fibration is preserved by
the group U (1, n), which acts on Ein2n as a subgroup of O(2, 2n). If Z
denotes the center of U (1, n) (homotheties by complex numbers of modulus
1), then the fibers of f are exactly the orbits of Z on Ein2n . These orbits
are lightlike geodesics.

Proposition 7. If Γ ∈ U (1, n) is a discrete group, whose projection Γ̂ on

P U (1, n) acts properly discontinuously on Ω̂ ⊂ S2n−1 , then Γ is a Kleinian
group of Ein2n and acts properly discontinuously on Ω = f −1 (Ω̂). If Ĝ acts
with compact quotient on Ω̂, so does Γ on Ω.

Remark 5. The group P U (1, n) acting on S2n−1 is a convergence group,

and there is a good notion of limit set for a discrete group Ĝ as above (see
for example [A]). In fact, it is not difficult to check that the Lorentzian
Kleinian groups Γ built as in proposition 7 are of the first type. Their limit
set is just the preimage by f of the limit set Λ̂Γ̂ of Γ̂ on S2n−1 .

To illustrate this case, let us mention the two following examples:

Example 1.

16
We write each z ∈ Cn+1 as z = (x, y) with x and y in Rn . We identify the
real hyperbolic space HnR with the set of points (x, 0) with −x21 + x22 + ... +
x2n+1 = −1 and x1 > 0. If (x, y) is moreover in the unit tangent bundle of
HnR , it satisfies the two extra equations:
−x1 y1 + x2 y2 + ... + xn yn = 0
−y12 + y22 + ... + yn+1
2 =1
Projectivising, we get an open subset Ω̂ ⊂ S2n−1 . In fact Ω̂ is precisely
S2n−1 minus a (n − 1)-dimensional sphere Σ (the projection on S2n−1 of the
set {z = (x, 0)| − x21 + x22 + ... + x2n+1 = 0}).
Now, the subgroup G = O(1, n) of real matrices in U (1, n) acts on S2n−1 ,
and preserves Ω̂. Identifying Ω̂ with T 1 HnR , we get that G acts properly and
transitively on Ω̂. As a consequence, we have the following:
Fact 1. Any discrete group Γ in O(1, n) acts properly discontinuously on
Ω̂. Seen as a subgroup of O(2, 2n) it yields a Kleinian group acting on Ein2n .

The Kleinian manifold Ω/Γ obtained in this way are circle bundles over
T 1 (N ), where N is the hyperbolic manifold HnR /Γ.

Example 2.
Inside U (1, n), there is a group G isomorphic to the Heisenberg group of
dimension 2n − 1. The group G fixes a point p∞ on S2n−1 and acts simply
transitively on the complement of this point. By proposition 7, any discrete
group in G will yield a Lorentzian Kleinian group, acting properly on the
complement of a lightlike geodesic. The Kleinian manifolds obtained in this
way will be circle bundles over nilmanifolds.

5.3 Subgroups of O(1, r) × O(1, s)

We still endow R2,n with the quadratic form q 2,n (x) = −2x1 xn+2 +2x2 xn+1 +
x23 + ... + x2n , and we consider an orthogonal splitting R2,n = E1 + E2 with
E1 and E2 two spaces of signature (1, r) and (1, s) respectively (r 6= 0, s 6= 0
and r + s = n). We suppose also r ≤ s. For example, we take E1 =
(e1 , e3 , ..., es , en+2 ) and E2 = (e2 , er+1 , ..., en+1 ). The subgroup G of O(2, n)
preserving this splitting is isomorphic to the product O(1, r)×O(1, s). Before
describing some examples of Kleinian groups in G, let us say a few words
about the geometric meaning of this splitting on Einn .

Lemma 3. We can write Einn as a union Ω1 ∪Ω2 ∪Σ. The set Ω1 (resp. Ω2 )
is open, G-invariant, homogeneous under the action of G, and conformally
equivalent to the product dSr × Hs (resp. Hr × dSs ). Σ is a singular,
degenerate G-invariant hypersurface.

Proof : We call π1 and π2 the projections of R2,n on E1 and E2 respectively.

The projection of vectors u = (v, w) of R2,n for which both v = π1 (u) and

17
w = π2 (u) are isotropic gives the hypersurface Σ. We will say more about
it later.
The vectors u = (v, w) for which neither v nor w is isotropic are of two
kinds.
- Those for which q 2,n (v) > 0.
Since we work projectively, we can suppose that q 2,n (v) = 1 and q 2,n (w) =
−1. In a further quotient by −Id, these vectors project on the product
dSr × Hs . They constitute the open set Ω1 .
- Those for which q 2,n (v) < 0.
These vectors project on a product Hr × dSs , and constitute the open set
Ω2 .


The hypersurface Σ can be seen as the conformal boundary of the spaces

dSr × Hs and Hr × dSs . Let us describe it more precisely. The isotropic
vectors (v, w) of R2,n for which v and w are isotropic split themselves into
two sets. Those for which either v or w is zero. Their projectivisation gives
two Riemannian spheres Σ1 and Σ2 of dimension (r − 1) and (s − 1) respec-
tively.
Those for which v and w are non zero project on the product of the projec-
tivisation of the lightcone of E1 by the lightcone of E2 , namely Sr−1 × C1,s .
So Σ minus Σ1 ∪ Σ2 has two connected components, each of which is diffeo-
morphic to Sr−1 × Ss−1 × R. One can check that Σ is obtained as the union
of the lightlike geodesics intersecting both Σ1 and Σ2 .
We now give some examples of Kleinian groups in G.

Example 3.
Let us take a discrete group Γ̂ inside O(1, r) and any representation ρ of Γ̂
inside O(1, s). We call Γρ = Graph(Γ̂, ρ) = {(γ̂, ρ(γ̂))|γ̂ ∈ Γ̂}. Then Γρ is a
Lorentzian Kleinian group of O(2, n). Indeed, its action on Ω2 = Hr × dSs
is clearly proper. Let us say a little bit more about the limit set of these
groups. We call ΛΓ̂ the limit set of the group Γ̂ on the sphere Σ1 .

Case a): ρ is injective with discrete image.  

γ̂k
A sequence (γk ) of Γρ can be written as a matrix . If (γk )
ρ(γ̂k )
tends simply to infinity, so does the sequence (γ̂k ) (resp. ρ(γ̂k )) in O(1, r)
(resp. in O(1, s)). We thus see that (γk ) has either mixed or balanced dis-
tortions. In particular, the group Γρ is always of the first type in this case.
The attracting and repelling circles of (γk ) can be decribe as follows. Since
the sequence (γ̂k ) (resp. ρ(γ̂k )) tends simply to infinity in O(1, r) (resp.
O(1, s)), it has two attracting and repelling poles p+ (γ̂k ) and p− (γ̂k ) (resp.
p+ (ρ(γ̂k )) and p− (ρ(γ̂k ))) on Σ1 (resp. Σ2 ). Then ∆+ (γk ) (resp. ∆− (γk ))
is simply the lightlike geodesic of Einn joining p+ (γ̂k ) and p+ (ρ(γ̂k )) (resp.

18
p− (γ̂k ) and p− (ρ(γ̂k ))). In particular, the limit set ΛΓρ is a closed subset of
Σ (strictly included in Σ if ΛΓ̂ 6= Σ1 ).
An interesting subcase arises when we take for Γ̂ a cocompact lattice in
O(1, 2), and a quasi-fuchsian representation ρ : Γ̂ → O(1, s) (s ≥ 2). The
limit set of ρ(Γ̂) on Σ2 is a topological circle, and we get for the limit set
ΛΓρ a topological torus. One can prove moreover (but we don’t do it here)
that the action of Γρ is cocompact on the complement of its limit set.

Cas b): ρ is not injective with discrete image.

In this case, there is a sequence (γk ) tending simply to infinity in Γρ such
that ρ(γ̂k ) is bounded. Such a sequence (γk ) has bounded distortion and the
group Γρ is of the second type. The attracting and repelling poles p+ (γk )
and p− (γk ) are both on Σ1 . In fact they are the attracting and repelling
poles of (γ̂k ) (acting as a sequence of O(1, r) on Σ1 ). In this case, the limit
set ΛΓρ is just the union of lightcones with vertex on ΛΓ̂ .

6 About Klein’s combination theorem

Until now, the examples of Kleinian groups we gave could not appear com-
pletely satisfactory, since they arise from geometrical contexts such as Loren-
tzian spaces with constant curvature or flat CR-geometry, and in some way
are not “typical” of conformally flat Lorentzian geometry. For example, we
still don’t have examples of Zariski dense Kleinian groups on Einn . One way
to construct other classes of examples, is to combine two existing Lorentzian
Kleinian groups to get a third one. In the theory of Kleinian groups on the
sphere, this kind of construction is achieved thanks to the celebrated Klein’s
combination theorem ([A] [Ma]). We state now a generalized version of this
theorem. Before this, we need the:
Definition 5. Let X be a manifold. A Kleinian group on X is a dis-
crete subgroup of diffeomorphisms Γ acting properly discontinuously on some
nonempty open set Ω ⊂ X. We say that an open set D ⊂ Ω is a fundamen-
tal domain for the action of Γ S
on Ω if D does not contain two points of the
same Γ-orbit and if moreover γ∈Γ γ(D) = Ω.
Notation 2. For any subset D of the manifold X, we call Ext(D) the
complement of D in X.
Theorem 5 (Klein). Let Γi (i = 1, ..., m) a finite family of Kleinian groups
on a compact connected manifold X. We suppose that each Γi acts cocom-
pactly on some open subset Ωi of X, with fundamental domain DT i . We as-
sume moreover that for each i 6= j, Ext(Di ) ⊂ Dj , and that D = m i=1 Di 6=
∅. Then:
(i) The group Γ generated by the Γi ’s is isomorphic to the free product
Γ1 ∗ ... ∗ Γm .

19
 S
(ii) The group Γ is Kleinian. More precisely, Ω = γ∈Γ γ(D) is an open
subset of X, and Γ acts properly discontinuously and cocompactly on Ω, with
fundamental domain D.

Proof : We do the proof for two groups Γ1 and Γ2 , the final result being then
obtained by induction. Let γ = γs γs−1 ...γ2 γ1 be a word of Γ such that γi ∈
Gji (ji ∈ {1, 2}) and ji 6= ji+1 . Then, the first condition on the fundamental
domains yields the inclusions γs γs−1 ...γ2 γ1 (D) ⊂ γs γs−1 ...γ2 (Ext(Dj1 )) ⊂
... ⊂ γs (Ext(Djs−1 )) ⊂ Ext(Djs ). So, for any non trivial reduced g, γ(D) ∩
D = ∅. This proves that γ can’t be the identity and the point (i) follows.
In the same way, we prove that γ(D) ∩ D = ∅ as soon as s > 1. Since D is
compact in Ω1 and Ω2 and the action of Γ1 and Γ2 is proper, we get the:

Lemma 4. The intersection γ(D) ∩ D is empty for all but a finite number
of γ’s.

Lemma 5. There is a finite family γ1 , ..., γs of elements of Γ such that

D ∪ γ1 (D) ∪ ... ∪ γm (D) contains D in its interior.

Proof : We choose some open neighbourhood U1 of ∂D1 such that U1 ⊂ Ω1

and U 1 is a compact subset of Ω1 . Since D1 is a fundamental domain of Γ1 ,
for each x ∈ U1 , there exists a γx ∈ Γ1 such that x ∈ γx (D1 ). But since the
action of Γ1 is proper γ(D1 ) ∩ U1 is nonempty only for a finite number of
(1) (1) (1)
elements γ1 , ..., γs of Γ1 . Thus D1 ∪ U1 is included in D1 ∪ γ1 (D1 ) ∪ ... ∪
(1) (1) (1)
γs (D1 ) and D1 is contained in the interior of D1 ∪ γ1 (D1 ) ∪ ... ∪ γs (D1 ).
But if D10 = D1 \K, where K is a compact subset of D1 , then we also have
0 0 (1) 0 (1) 0
D1 ∪U1 ⊂ D1 ∪γ1 (D1 )∪...∪γs (D1 ). In particular, when K is the exterior
(1) (1)
of D2 , we get that D ∪ U1 ⊂ D ∪ γ1 (D) ∪ ... ∪ γs (D). Now, we can apply
the same argument for a neighbourhood U2 of ∂D2 in Ω2 . We get a finite
(2) (2) (2) (2)
family γ1 , ..., γt of Γ2 such that D ∪ U2 ⊂ D ∪ γ1 (D) ∪ ... ∪ γt (D).
(1) (2)
Setting m = s + t, γi = γi for i = 1, ..., s and γs+i = γi for i = 1, ..., t, we
get the lemma. 
S
As a consequence of this lemma, we get that the set Ω = γ∈Γ γ(D) is an
open set.
It remains to prove that the action of Γ on Ω is proper. Indeed, since Γ is
not a priori a convergence group, the fact that Γ acts discontinuously on Ω
no longer ensures that the action is proper. That is why our assumptions (in
particular the assumption of cocompactness) are stronger as for the classical
Klein’s theorem on the sphere.
Suppose, on the contrary, that there is a sequence (xi ) of Ω converging
to x∞ ∈ Ω, and a sequence (γi ) tending to infinity in Γ, such that yi = γi (xi )
converges to y∞ ∈ Ω. We can assume that x∞ ∈ D. On the other hand, by
definition of Ω, there is a γ0 such that y∞ ∈ γ0 (D). The lemma 5 ensures

20
that for i sufficiently large, xi must be in D ∪ γi1 (D) ∪ ... ∪ γim (D), and yi
in γ0 (D) ∪ γ0 γi1 (D) ∪ ... ∪ γ0 γim (D)). But then, lemma 4 implies that the
sequence (γi ) takes its values in a finite set, a contradiction with the fact
that (γi ) tends to infinity in Γ.


We would like to apply the theorem above to combine Lorentzian Kleinian

groups. Notice that for two Kleinian groups, the condition Ext(D1 ) ⊂ D2
implies ∂Ω1 ⊂ D2 and ∂Ω2 ⊂ D1 . Together with lemma 2, we get that if two
cocompact Lorentzian Kleinian groups can be combined, their fundamen-
tal domains have to contain a lightlike geodesic (in particular, no Kleinian
group uniformizing a manifold with constant curvature can be combined
with another Kleinian group). It turns out that this obstruction is the only
one which forbids combining two Lorentzian Kleinian groups, as shown by
theorem 2, which we now prove.

6.1 Proof of Theorem 2

We choose ∆1 ⊂ D1 and ∆2 ⊂ D2 two lightlike geodesics. Since D1 and
D2 are open, they contain not only one, but in fact infinitely many lightlike
geodesics, so that we can moreover choose ∆1 and ∆2 disjoint. We now
state the:

Lemma 6. Given ∆1 and ∆2 two disjoint lightlike geodesics of Einn , there

exists g ∈ Conf(Einn ) such that (g k ) has mixed distortions and admits ∆1
and ∆2 as attracting and repelling circles.

Proof : The geodesic ∆1 (resp. ∆2 ) is the projection on Einn of a 2-

plane (e01 , e02 ) (resp. (e03 , e04 )) of R2,n . We choose moreover e03 and e04
such that q 2,n (e01 , e03 ) = −2 and q 2,n (e02 , e04 ) = 2. The q 2,n -orthogonal F
to (e01 , e02 , e03 , e04 ) has Riemannian signature and we denote by e05 , ..., e0n+2 one
of its orthonormal basis. Then we consider some element g of O(2, n), which
writes in the base (e01 , ..., e0n+2 ):
 λ 
e
 eµ 
−λ
 

 e 

e−µ
g=
 


 1 

 .. 
 . 
1
If we choose λ > µ > 0, then it is clear that (g k ) has mixed distortions
with ∆+ = ∆1 and ∆− = ∆2 . 

21
We now take some g ∈ Conf(Einn ) as in the lemma above. Let us choose
V1 (resp. V2 ) an open tubular neighbourhood of ∆1 (resp. ∆2 ) such that
V 1 ⊂ D1 (resp. V 2 ⊂ D2 ). The complement of Vi (i = 1, 2) in Einn is
denoted by Ext(Vi ). It follows from proposition 5 (points (i) and (ii)) that
the set dynamically associated to Ext(V2 ) with respect to (g k ) is included in
∆+ . Since Ext(V2 ) contains a lightlike geodesic, it is exactly ∆+ . Hence, for
k0 sufficiently large, g k0 (Ext(V2 )) ⊂ V1 . We call Γ02 = g k0 Γ2 g −k0 . The group
Γ02 is a cocompact Lorentzian Kleinian group, with fundamental domain
D20 = g k0 (D2 ). But g k0 (D2 ) contains g k0 (Int(V2 )), and as we just saw,
Ext(V1 ) ⊂ g k0 (Int(V2 )). So Ext(D20 ) ⊂ D1 . We can then apply theorem
5, and we get that the group generated by Γ02 and Γ1 is still Kleinian,
cocompact, and isomorphic to Γ1 ∗ Γ02 , i.e Γ1 ∗ Γ2 .
Example 4
All the cocompact Lorentzian Kleinian groups of the examples 1 and 2 of
section 5 satisfy the hypothesis of theorem 2. This is also the case of most
instances of example 3, when ρ is injective with discrete image. Thus, such
groups can be combined and give new examples. Notice that in the proof of
theorem 2, the gluing element g can be chosen in many ways. In particular,
starting from two groups of the examples 1, 2 or 3, suitable choices of g will
give combined groups which are Zariski dense in O(2, n).

6.2 Lorentzian surgery

Theorem 2 is in fact the group theoretical aspect of a slightly more general
process of conformal Lorentzian surgery.
Let M1 and M2 be two conformally flat Lorentzian manifolds (we don’t
make any compactness assumption). Suppose that M1 contains a closed
lightlike geodesic ∆1 , admitting some open neighbourhood U1 which em-
beds conformally, via a certain embedding φ1 , into Einn . Suppose moreover
that the same property is satisfied by M2 , for a closed lightlike geodesic
∆2 , an open neighbourhood U2 , and a conformal embedding φ2 . We can
suppose that φ1 (∆1 ) and φ2 (∆2 ) are disjoint in Einn . By lemma 6, φ1 (∆1 )
and φ2 (∆2 ) are the attracting and repelling circles of some element g ∈
Conf(Einn ). As in the proof of theorem 2, there exist two open neighbour-
hoods V1 and V2 of ∆1 and ∆2 respectively, such that V1 ⊂ U1 , V2 ⊂ U2 ,
and g(Ext(φ2 (V2 ))) = φ1 (V1 ). In particular g(∂(φ2 (V2 ))) = ∂(φ1 (V1 )) (re-
call that ∂ denotes the boundary). so, the element g provides a gluing map
f between ∂V1 and ∂V2 . We denote by Ṁ1 (resp. Ṁ2 ) the manifold M1
(resp. M2 ) with V1 (resp. V2 ) removed. We call M = Ṁ1 ]f Ṁ2 the manifold
obtained from Ṁ1 ∪ Ṁ2 , after identification of ∂V1 and ∂V2 by means of the
map f . Since g ∈ Conf(Einn ), the “surgered manifold” M is still endowed
with a conformally flat Lorentzian structure. Theorem 2 ensures that if one
starts with two compact Kleinian structures M1 and M2 , the conformally
flat structure on Ṁ1 ]f Ṁ2 is still Kleinian.

22
Remark 6. This surgery process is reminiscent of Kulkarni’s construction
of a conformally flat Riemannian structure on the connected sum of two con-
formally flat Riemannian manifolds ([K1]). We don’t know if the connected
sum of two conformally flat Lorentzian manifolds can still be endowed with
a conformally flat Lorentzian structure.

7 Lorentzian Schottky groups

As an application of the former sections, we study here the Lorentzian Schot-
tky groups. These groups are interesting since we can completely determine
their limit set and the Kleinian manifolds they uniformize. Moreover, they
can be used to construct examples of conformally flat manifolds with some
peculiar properties (see [Fr2]).
Let us consider a family {(∆− + − +
1 , ∆1 ), ..., (∆g , ∆g )} of pairs of lightlike
geodesics in Einn . We suppose moreover that the ∆± i are all disjoint. By
lemma 6, there exists a family s1 ,..., sg of elements of Conf(Einn ) with mixed
dynamics such that the attracting and repelling circles of si are precisely ∆+ i
and ∆− i . Looking if necessary at suitable powers ski
i of si , we can find open
tubular neighbourhoods Ui± of the ∆± i such that:
±
(i) The U i are all disjoint.
+
(ii) si (Ext(Ui− )) = U i for all i = 1, ..., g.
Such a group Γ = hs1 , ..., sg i is called a Lorentzian Schottky group. Prop-
erties (i) and (ii) are classically known as ping-pong dynamics (see for ex-
ample [dlH]). For each i, the group hsi i acts properly cocompactly on the
open set Einn \{∆− +
i ∪ ∆i }, and a fundamental domain is just given by
+ − ±
Di = Einn \{U i ∪ U i }. Now, since theTU i are disjoint, we get that
Ext(Di ) ⊂ Dj for all i 6= j. If we call D = gi=1 Di , it is clear that D 6= ∅.
We then apply theorem 5 to obtain:

Proposition 8. A Lorentzian Schottky group Γ = hs1 , ..., sg i is a free group

of Conf(Ein
S n ). Moreover, Γ is Kleinian: it acts properly and cocompactly
on ΩT= γ∈Γ γ(D). A fundamental domain for this action is given by
D = gi=1 Di .

We are now going to describe Ω, and its complement Λ ⊂ Einn more

precisely.
Let us recall that in a finitely generated free group, each element γ can be
written in an unique way as a reduced word in the generators. We denote
by |γ| the length of this word. Let us also recall that we can define the
boundary ∂Γ of Γ as the set of totally reduced words of infinite length. So,
the elements of the boundary can be wrotten si11 ....sikk ... with j ∈ {±1} and
ij j 6= −ij+1 j+1 for all j ≥ 1. Since we supposed g ≥ 2, the boundary ∂Γ
is a compact metrizable space, homeomorphic to a Cantor set (see [GdlH]).

23
 S
For each k ∈ N, we call Fk = |γ|≤k γ(D), with the convention F0 =
S
T not difficult to check that Fk−1 ⊂ Fk , and Ω = k∈N Fk . So,
D. It’s
Λ = k∈NT Ext(Fk ). For each k, we set Λk = Ext(Fk ), and thus, we also
have Λ = k∈N Λk . The set Λk is a disjoint union of exactly 2g.(2g − 1)k
connected components, in one to one correspondence with the words of
k+1
length k + 1 in Γ. For example, to the word si11 ...sik+1 corresponds the

component si11 ...sikk (U ik+1
k+1
) of Λk . We can now state:
Lemma 7. There is an homeomorphism K between the boundary ∂Γ and the
space of connected components of Λ (endowed with the Hausdorff topology
for the compact subsets of Einn ).

Proof : Let γ∞ = si11 ....sikk ... be an element of ∂Γ. We call γk = si11 ....sikk and
k−1 
we look at the decreasing sequence of compact subsets K(γk ) = si11 ....sik−1 (U ikk ).
This decreasing sequence of compact sets tends to a limit compact set K(γ∞ )
for the Hausdorff topology. Since the Ui± are connected, so are the K(γk ),
and K(γ∞ ) is itself connected. Let us remark that if γ∞ and γ∞ 0 are distinct
0
in ∂Γ, then K(γk ) and K(γk ) are disjoint for k large (they represent two
distinct components of Λk ), so that K(γ∞ ) and T K(γ∞0 ) are disjoint.

Reciprocally, choose x∞ ∈ Λ. Since Λ = k∈N Λk with Λk+1 ⊂ Λk ,

x∞ must be an element of some connected component Ck ⊂ Λk for each
k. Moreover Ck+1 ⊂ Ck . But Ck is then a decreasing sequence of compact
k−1 
subsets of the form si11 ....sik−1 (U ikk ), and thus converges to a limit compact
set K(γ∞ ) for γ∞ = si11 ....sikk ....
We have proved that the mapping K between ∂Γ and the set of con-
nected components of Λ is a bijection. It remains to prove that it is an
homeomorphism, and for this, it is sufficient to show that K is continuous.
(n)
Let us consider a sequence γ∞ of elements of Γ, converging to some γ∞ .
It means that there is a sequence (rn ) of integers which tends to infinity,
(n)
such that γ∞ and γ∞ have the same rn first letters. For each n ∈ N,
(n) (n) (n)
K(γ∞ ) is a decreasing sequence of compact sets Ck , where each Ck is
a connected component of Λk . On the other hand, K(γ∞ ) is the limit of a
decreasing sequence of Ck , where each Ck is a connected component of Λk .
(n) (n)
Since γ∞ and γ∞ have the same rn first letters, we have Crn −1 = Crn −1 for
(n)
all n. Thus, the limit, as n tends to infinity, of Crn −1 is K(γ∞ ). But since
(n) (n) (n)
K(γ∞ ) ⊂ Crn −1 , we get that limn→∞ K(γ∞ ) = K(γ∞ ) and we are done.


The next step is the:

Lemma 8. The connected components of Λ are lightlike geodesics.

Proof : Let us consider γ∞ = si11 ...sikk ... in the boundary of Γ. We know that
k−1 
K(γ∞ ) is the limit of the sequence si11 ...sik−1 (U ikk ). Since the sequence is

24
decreasing, the limit remains the same if we consider a subsequence. Thus,
we can make the extra assumption that K(γ∞ ) is the limit of a sequence
j
γk (U j00 ), such that (γk ) tends simply to infinity and the first and last letters
j
of γk are always the same, namely si11 and sj11 . Let us precise that j1 j1 6=
−j0 j0 . We are going to discuss the different possible dynamics for (γk ), and
we first prove that (γk ) can’t have bounded distortion.
Suppose that it is the case. We call p+ (resp. p− ) and C + (resp. C − ) the
attracting (resp. repelling) pole and cone of (γk ). If x is a point of D, then
−j
for all k ∈ N, γk (x) ∈ Ui11 and γk −1 (x) ∈ Uj1 1 . So, we must have p+ ∈ Ui11
−j j
and p− ∈ Uj1 1 . In particular, p− is not in Uj0 0 . On the other hand,
this is a general fact that in Einn , any lightlike cone meets any lightlike
geodesic (just because degenerate hyperplanes always meet null 2-planes
j j
in R2,n ). In particular, the cone C − meets ∆j00 , and thus Uj0 0 . We call
j j j
Vj0 0 = C − ∩ Uj0 0 . Since Uj0 0 does not contain p− , we infer from proposition
j j
4 (points (i) and (ii)) that K(γ∞ ) = D(γk ) (V j00 ). More precisely, if V̂j0 0 is
j
the set of lightlike geodesics of C − meeting Vj0 0 , then K(γ∞ ) is the closure
j
of the union the lightlike geodesics of γ̂∞ (V̂j0 0 ) (see proposition 4 for the
notation γ̂∞ ). In particular, K(γ∞ ) contains a lightlike geodesic. Now, some
j
lightlike geodesic of C − does not meet Vj0 0 . Indeed, if it is not the case,
the point (ii) of proposition 4 ensures that K(γ∞ ) = C − . But if we take
γ∞0 6= γ , K(γ 0 ) contains some lightlike geodesic by the remark above, and
∞ ∞
since any lightlike geodesic meets C − , we get a contradiction with the fact
that K(γ∞ ) and K(γ∞ ) have to be disjoint.
j −j
Now, let us perturb slightly the sets Uj0 0 and Uj0 0 into some sets Uj00 j0
and Uj00 −j0 , in order to get another fundamental domain D0 , very close to
0
D. Since it isS very near D,SD is included in some Fk for k sufficiently
0
large, and so γ∈Γ γ(D ) = γ∈Γ γ(D). We prove as above that the limit
of the compact sets γk (Uj00 j0 ) is still a connected component of Λ, and
consequentely of the form K(γ∞ 0 ). We just saw that some lightlike geodesics
 j
of C − don’t meet V j00 , so that V̂j0 0 is not the whole Sn−2 . It is thus possible
j

j j
to choose U 0 j0 in such a way that some points of Vˆ0 0 are not in V̂ 0 . But
j0 j0 j0
then, K(γ∞ 0 ) and K(γ ) will be two different components, hence disjoint.
∞
j
On the other hand, since the intersection of Uj0 0 and Uj00 j0 is not empty
j 0 ) must have some common points. We
(∆j00 is inside), K(γ∞ ) and K(γ∞
thus get a contradiction.
It remains to deal with the case where (γk ) has mixed or balanced dis-
tortions. Once again, if x is a point of D then for all k ∈ N, γk (x) ∈ Ui11 and
−j1
γk −1 (x) ∈ Uj1 . Hence, the attracting circle ∆+ is in Ui11 and the repelling
−j j
one ∆− is in Uj1 1 . In particular U j00 does not meet ∆− . We infer from
j j
proposition 5 and proposition 3 that limk→∞ γk (U j00 ) ⊂ ∆+ , but since U j00

25
 j
contains a lightlike geodesic, we have the equality limk→∞ γk (U j00 ) = ∆+ .
We finally obtain that K(γ∞ ) = ∆+ .


7.1 Proof of Theorem 3

We begin by proving that the group Γ is of the first type. Suppose on the
contrary that there is some sequence (γk ) in Γ with bounded distortion.
Then D meets the repelling cone C − . Indeed, if not, C − would be included
in some Ui± , for example U1+ . But since ∆− −
1 meets C , the intersection
− +
between ∆1 and U1 would be non empty, a contradiction. By the point
(ii) of proposition 4, limk→∞ γk (D) is a compact subset containing infinitely
many lightlike geodesics. But limk→∞ γk (D) is also a connected subset of
Λ. This contradict the fact that the connected components of Λ are lightlike
geodesics.

Now, we claim that the equality ΛΓ = Λ holds. Indeed, for any sequence
(γk ) of Γ tending simply to infinity, (γk ) tends to ∆+ (γk ). We thus see
that ΛΓ ⊂ Λ. Now, it is a general fact that if a group Γ acts properly co-
compactly on some open set Ω, it can’t act properly on some open set Ω0
strictly containing Ω. So, Ω can’t be strictly contained in Einn \ΛΓ and we
get ΛΓ = Λ.

Thanks to the homeomorphism K, we get that since the action of Γ on

its boundary is minimal (see for instance [GdlH]), the action of Γ on the
space of lightlike geodesics of ΛΓ is also minimal.
We now prove that ΛΓ is the product of RP 1 with a Cantor set. The
space Einn is the quotient of S1 × Sn−1 by the product of antipodal maps,
so that there is a fibration f : Einn → RP 1 . The fibers of f are conformal
Riemannian spheres of codimension one. In the projective model, they are
obtained as the projection of the intersection between C 2,n and some hyper-
planes P ⊂ R2,n of Lorentzian signature. As a consequence, any lightlike
geodesic is transverse to any fiber of f . Let us choose a fiber F0 above a
point t0 of RP 1 . From lemmas 7 and 8, Λ (and thus ΛΓ ) is transverse to
F0 and intersects it along a Cantor set C. For each x ∈ C, we call x(t) the
unique element of f −1 (t) ∩ ΛΓ such that x and x(t) are on the same lightlike
geodesic of Λ. Then, the lemma 7 ensures that the following mapping is an
homeomorphism.
RP 1 × C → Λ
(t, x) 7→ x(t)
The point (ii) is then proved.
Thanks to the homeomorphism K, we get that since the action of Γ on
its boundary is minimal (see for instance [GdlH]), the action of Γ on the
space of lightlike geodesics of ΛΓ is also minimal, what proves (iii).

26
 For the proof of (iv), we refer to the theorem 5 of [Fr2] (in fact, in [Fr2],
we considered only particular cases of Schottky groups, but the proof of
theorem 5 includes the general case).

Aknowledgments: I would like to thank Thierry Barbot for his useful

remarks on this text, the ACI ”Structures géométriques et trous noirs”, as
well as the referee, who made a lot of valuable suggestions to improve the
original version. Thank you also to the Institüt für Mathematik of Humboldt
University in Berlin, where part of this work was done in autumn 2003.

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Charles FRANCES
Laboratoire de Topologie et Dynamique
Université de Paris-Sud, Bât. 430
91405 ORSAY
FRANCE
email: Charles.Frances@math.u-psud.fr

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