We describe the orbits of the irreducible action of PSL(2, R) on the
3-dimensional Einstein universe Ein 1,2. This work completes the study in [2],
and is one element of the classification of cohomo-geneity one actions on Ein
1,2 ([5])

Hassani, Masoud

English

arXiv

We describe the orbits of the irreducible action of PSL(2,R) on the 3-dimensional Einstein universe Ein 1,2. This work completes the study in [2], and is one element of the classification of cohomo-geneity one actions on Ein 1,2 ([5])

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ON THE IRREDUCIBLE ACTION OF PSL(2,R) ON
THE 3-DIMENSIONAL EINSTEIN UNIVERSE
Masoud Hassani
To cite this version:
Masoud Hassani. ON THE IRREDUCIBLE ACTION OF PSL(2,R) ON THE 3-DIMENSIONAL
EINSTEIN UNIVERSE. 2017. ￿hal-01547611￿
ON THE IRREDUCIBLE ACTION OF PSL(2,R) ON THE 3-DIMENSIONAL
EINSTEIN UNIVERSE.
MASOUD HASSANI
Abstract. We describe the orbits of the irreducible action of PSL(2,R) on the 3-dimensional Einstein
universe Ein1,2. This work completes the study in [2], and is one element of the classification of cohomo-
geneity one actions on Ein1,2 ([5]).
1. Introduction
1.1. Einstein universe. Let R2,n+1 denote a (n + 3)-dimensional real vector space equipped with a
non-degenerate symmetric bilinear form q with signature (2, n+ 1). The nullcone of R2,n+1 is
N2,n+1 = {v ∈ R2,n+1 \ {0} : q(v) = 0}.
The (n+ 1)-dimensional Einstein universe Ein1,n is the image of the nullcone N2,n+1 under the projec-
tivization:
P : R2,n+1 \ {0} −→ RPn+2.
The degenerate metric on N2,n+1 induces a O(2, n+ 1)-invariant conformal Lorentzian structure on Ein-
stein universe. The group of conformal transformations on Ein1,n is O(2, n+ 1) [4].
A lightlike geodesic in Einstein universe is a photon. A photon is the projectivisation of an isotropic
2-plane in R2,n+1. The set of photons through a point p ∈ Ein1,n denoted by L(p) is the lightcone at p.
The complement of a lightcone L(p) in Einstein universe is the Minkowski patch at p and we denote
it by Mink(p). A Minkowski patch is conformally equivalent to the (n + 1)-dimensional Minkoski space
E1,n [1].
The complement of the Einstein universe in RPn+2 has two connected components: the (n + 2)-
dimensional Anti de-Sitter space AdS1,n+1 and the generalized hyperbolic space H2,n: the first (respec-
tively the second) is the projection of the domain R2,n+1 defined by {q < 0} (respectively {q > 0}).
An immersed submanifold S of AdS1,n+1 or H2,n is of signature (p, q, r) (respectively Ein1,n) if the
restriction of the ambient pseudo-Riemmanian metric (respectively the conformal Lorentzian metric) is
of signature (p, q, r), meaning that the radical has dimension r, and that maximal definite negative and
positive subspaces have dimensions p and q, respectively. If S is nondegenerate, we forgot r and simply
denote its signature by (p, q).
1.2. The irreducible representation of PSL(2,R). A subgroup of O(2, n + 1) is irreducible if it
preserves no proper subspace of R2,n+1. By [3, Theorem 1], up to conjugacy, SO◦(1, 2) ' PSL(2,R) is
the only irreducible connected Lie subgroup of O(2, 3).
This work is funded by the French Ministry of Foreign Affairs, through Campus France. Grant number 878286E.
1
2 MASOUD HASSANI
On the other hand, for every integer n, it is well known that, up to isomorphism, there is only one
n-dimensional irreducible representation of PSL(2,R). For n = 5, this representation is the natural action
of PSL(2,R) on the vector space V = R4[X,Y ] of homogeneous polynomials of degree 4 in two variables
X and Y . This action preserves the following quadratic form
q(a4X
4 + a3X
3Y + a2X
2Y 2 + a1XY
3 + a0Y
4) = 2a4a0 −
1
2
a1a3 +
1
6
a22.
The quadratic form q is nondegenerate and has signature (2, 3). This induces an irreducible representation
PSL(2,R)→ O(2, 3) [2].
Theorem 1.1. The irreducible action of PSL(2,R) on the 3-dimensional Einstein universe Ein1,2 admits
three orbits:
• An 1-dimensional lightlike orbit, i.e. of signature (0, 0, 1)
• A 2-dimensional orbit of signature (0, 1, 1),
• An open orbit (hence of signature (1, 2)) on which the action is free.
The 1-dimensional orbit is lightlike, homeomorphic to RP1, but not a photon. The union of the 1-
dimensional orbit and the 2-dimensional orbit is an algebraic surface, whose singular locus is precisely the
1-dimensional orbit. It is the union of all projective lines tangent to the 1-dimensional orbit. Figure 1
describes a part of the 1 and 2-dimensional orbits in the Minkowski patch Mink(Y 4).
Figure 1. Two partial views of the intersection of the 1 and 2-dimensional orbits in
Einstein universe with Mink(Y 4). Red: Part of the 1-dimensional orbit in Minkowski
patch. Green: Part of the 2-dimensional orbit in Minkowski patch.
We will also describe the actions on Anti de-Sitter space and the generalized hyperbolic space H2,2:
Theorem 1.2. The orbits of PSL(2,R) in the Anti de-sitter component AdS1,3 are Lorentzian, i.e. of
signature (1, 2). They are the leaves of a codimension 1 foliation. In addition, PSL(2,R) induces three
types of orbits in H2,2: a 2-dimensional spacelike orbit (of signature (2, 0)) homeomorphic to the hyperbolic
plane H2, a 2-dimensional Lorentzian orbit (i.e., of signature (1, 1)) homeomorphic to the de-Sitter space
dS1,1, and four kinds of 3-dimensional orbits where the action is free:
• one-parameter family of orbits of signature (2, 1) consisting of elements with four distinct non-real
roots,
ON THE IRREDUCIBLE ACTION OF PSL(2,R) ON THE 3-DIMENSIONAL EINSTEIN UNIVERSE. 3
• one-parameter family of Lorentzian (i.e. of signature (1, 2)) orbits consisting of elements with
four distinct real roots,
• two orbits of signature (1, 1, 1),
• one-parameter family of Lorentzian (i.e. of signature (1, 2)) orbits consisting of elements with two
distinct real roots, and a complex root z in H2 making an angle θ smaller than 5π/6 with the two
real roots.
Remark 1.3. F. Fillastre indicated to us an alternative description for the last case stated in Theorem 1.2:
these orbits correspond to polynomials whose roots in CP1 are ideal vertexes of regular ideal tetraedra in
H3.
2. Proofs of the Theorems
Let f be an element in V. We consider it as a polynomial function from C2 into C. Actually, by
specifying Y = 1, we consider f as a polynomial of degree at most 4. Such a polynomial is determined,
up to a scalar, by its roots z1, z2, z3, z4 in CP1 (some of these roots can be ∞ if f can be divided by Y ).
It provides a natural identification between P(V) and the set ĈP
1
4 made of 4-tuples (up to permutation)
(z1, z2, z3, z4) of CP1 such that if some zi is not in RP1, then its conjugate z̄i is one of the zj ’s. This
identification is PSL(2,R)-equivariant, where the action of PSL(2,R) on ĈP
1
4 is simply the one induced
by the diagonal action on (CP1)4.
Actually, the complement of RP1 in CP1 is the union of the upper half-plane model H2 of the hyperbolic
plane, and the lower half-plane. We can represent every element of ĈP
1
4 by a 4-tuple (up to permutation)
(z1, z2, z3, z4) such that:
– either every zi lies in RP1,
– or z1, z2 lies in RP1, z3 lies in H2 and z4 = z̄3,
– or z1, z2 lies in H2 and z3 = z̄1, z4 = z̄2.
Theorems 1.1 and 1.2 will follow from the following proposition:
Proposition 2.1. Let [f ] be an element of P(V). Then:
• it lies in Ein1,2 if and only if it has a root of multiplicity at least 3, or two distinct real roots z1,
z2, and two complex roots z3, z4 = z̄3, with z3 in H2 and such that the angle at z3 between the
hyperbolic geodesic rays [z3, z1) and [z3, z2) is 5π/6,
• it lies in AdS1,3 if and only it has two distinct real roots z1, z2, and two complex roots z3, z4 = z̄3,
with z3 in H2 and such that the angle at z3 between the hyperbolic geodesic rays [z3, z1) and [z3, z2)
is > 5π/6,
• it lies in H2,2 if and only if it has no real roots, or four distinct real roots, or a root of multiplicity
exactly 2, or it has two distinct real roots z1, z2, and two complex roots z3, z4 = z̄3, with z3 in H2
and such that the angle at z3 between the hyperbolic geodesic rays [z3, z1) and [z3, z2) is < 5π/6.
Proof of Proposition 2.1. Assume that f has no real root. Hence we are in the situation where z1,
z2 lie in H2 and z3 = z̄1, z4 = z̄2. By applying a suitable element of PSL(2,R), we can assume z1 = i,
and z2 = ri for some r > 0. In other words, f is in the PSL(2,R)-orbit of (X2 + Y 2)(X2 + r2Y 2). The
value of q on this polynomial is 2× 1× r2 + 16 (1 + r
2)2 > 0, hence [f ] lies in H2,2.
Hence we can assume that f admits at least one root in RP1, and by applying a suitable element of
PSL(2,R), one can assume that this root is ∞, i.e. that f is a multiple of Y .
4 MASOUD HASSANI
We first consider the case where this real root has multiplicity at least 2:
f = Y 2(aX2 + bXY + cY 2)
Then, q(f) = 16a
2: it follows that if f has a root of multiplicity at least 3, it lies in Ein1,2, and if it has a
real root of mulitplicity 2, it lies in H2,2.
We assume from now that the real roots of f have multiplicity 1. Assume that all roots are real. Up
to PSL(2,R), one can assume that these roots are 0, 1, r and ∞ with 0 < r < 1.
f(X,Y ) = XY (X − Y )(X − rY ) = X3Y − (r + 1)X2Y 2 + rXY 3
Then, q(f) = − 12r +
1
6 (r + 1)
2 = 16 (r
2 − r + 1) > 0. Therefore f lies in H2,2 once more.
The only remaining case is the case where f has two distinct real roots, and two complex conjugate
roots z, z̄ with z ∈ H2. Up to PSL(2,R), one can assume that the real roots are 0, ∞, hence:
f(X,Y ) = XY (X − zY )(X − z̄Y ) = XY (X2 − 2|z| cos θXY + |z|2Y 2)
where z = |z|eiθ. Then:
q(f) =
2|z|2
3
(cos2 θ − 3
4
)
Hence f lies in Ein1,2 if and only if θ = π/6 or 5π/6. The proposition follows easily. 
Remark 2.2. In order to determine the signature of the orbits induced by PSL(2,R) in P(V), we consider
the tangent vectors induced by the action of 1-parameter subgroups of PSL(2,R). We denote by E, P and
H, the 1-parameter elliptic, parabolic and hyperbolic subgroups stabilizing i, ∞ and {0,∞}, respectively.
Proof of Theorem 1.1. It follows from Proposition 2.1 that there are precisely three PSL(2,R)-orbits
in Ein1,2:
– one orbit N comprising polynomials with a root of multiplicity 4, i.e. of the form [(sY − tX)4] with
s, t ∈ R. It is clearly 1-dimensional, and equivariantly homeomorphic to RP1 with the usual projective
action of PSL(2,R). Since ddt |t=0(Y − tX)
4 = −4XY 3 is a q-null vector, this orbit is lightlike (but cannot
be a photon since the action is irreducible),
– one orbit L comprising polynomials with a real root of multiplicity 3, and another real root. These
are the polynomials of the form [(sY − tX)3(s′Y − t′X)] with s, t, s′, t′ ∈ R. It is 2-dimensional, and it is
easy to see that it is the union of the projective lines tangent to N . The vectors tangent to L induced by
the 1-parameter subgroups P and E at [XY 3] ∈ L are vP = −Y 4 and vE = 3X2Y 2 + Y 4. Obviously, vP
is orthogonal to vE and vE + vP is spacelike. Hence L is of signature (0, 1, 1).
– one open orbit comprising polynomials admitting two distinct real roots and a root z in H2 making
an angle 5π/6 with the two real roots in ∂H2. The stabilizers of points in this orbit are trivial since an
isometry of H2 preserving a point in H2 and one point in ∂H2 is necessarily the identity. 
Proof of Theorem 1.2. According to Proposition 2.1, the polynomials in AdS1,3 have two distinct
real roots, and a complex root z in H2 making an angle θ greater than 5π/6 with the two real roots. It
follows that the action in AdS1,3 is free, and that the orbits are the level sets of the function θ. Suppose
that M is a PSL(2,R)-orbit in AdS1,3. There exists r ∈ R such that [f ] = [Y (X2 + Y 2)(X − rY )] ∈ M .
The orbit induced by the 1-parameter elliptic subgroup E at [f ] is
γ(t) =
[
(X2 +Y 2)
(
(sin t cos t− r sin2 t)X2− (sin t cos t+ r cos2 t)Y 2 + (cos2 t− sin2 t+ 2r sin t cos t)XY
)]
.
Then q(dγdt |t=0) = −2− 2r
2 < 0. This implies, as for any submanifold of a Lorentzian manifold admitting
a timelike vector, that M is Lorentzian, i.e., of signature (1, 2).
The case of H2,2 is the richest one. According to Proposition 2.1 there are four cases to consider:
ON THE IRREDUCIBLE ACTION OF PSL(2,R) ON THE 3-DIMENSIONAL EINSTEIN UNIVERSE. 5
• No real roots. Then f has two complex roots z1, z2 in H2 (and their conjugates). It corresponds to
two orbits: one orbit corresponding to the case z1 = z2: it is spacelike and has dimension 2. It is
the only maximal PSL(2,R)-invariant surface in H2,2 described in [2, Section 5.3]. The case z1 6= z2
provides a one-parameter family of 3-dimensional orbits on which the action is free (the parameter
being the hyperbolic distance between z1 and z2). One may assume that z1 = i and z2 = ri for
some r > 0. Denote by M the orbit induced by PSL(2,R) at [f ] = [(X2 + Y 2)(X2 + r2Y 2)]. The
vectors tangent to M at [f ] induced by the 1-parameter subgroups H, P and E are:
vH = −4X4+4r2Y 4, vP = −4X3Y − 2(r2 + 1)XY 3,
vE = 2(r
2 − 1)X3Y + 2(r2 − 1)XY 3,
respectively. The timelike vector vH is orthogonal to both vP and vE . It is easy to see that
the 2-plane generated by {vP , vE} is of signature (1, 1). Therefore, the tangent space T[f ]M is of
signature (2, 1).
• Four distinct real roots. This case provides a one-parameter family of 3-dimensional orbits on
which the action is free - the parameter being the cross-ratio between the roots in RP1. Denote
by M the PSL(2,R)-orbit at [f ] = [XY (X − Y )(X − rY )] (here as explained in the proof of
Proposition 2.1, we can restrict ourselves to the case 0 < r < 1). The vectors tangent to M at [f ]
induced by the 1-parameter subgroups H, P , and E are:
vH = −rY 4 + 2(r + 1)XY 3 − 3X2Y 2, vP = −2X3Y + 2rXY 3,
vE = X
4 − rY 4 + 3(r − 1)X2Y 2 + 2(r + 1)XY 3 − 2(r + 1)X3Y,
respectively. A vector x = avH + bvP + cvE is orthogonal to vP if and only if 2ra + b(r + 1) +
c(r + 1)2 = 0. Set a =
(
b(r + 1) + c(r + 1)2
)
/− 2r in
q(x) = 2ra2 +
3
2
b2 +
(7
2
(r2 + 1)− r
)
c2 + 2(r + 1)ab+ 2(r + 1)2 + ac(2r2 − r + 5).
Consider q(x) = 0 as a quadratic polynomial F in b. Since 0 < r < 1, the discriminant of F is
non-negative and it is positive when c 6= 0. Thus, the intersection of the orthogonal complement
of the spacelike vector vP with the tangent space T[f ]M is a 2-plane of signature (1, 1). This
implies that M is Lorentzian, i.e., of signature (1, 2).
• A root of multiplicity 2. Observe that if there is a non-real root of multiplicity 2, when we are in
the first ”no real root” case. Hence we consider here only the case where the root of multiplicity
2 lies in RP1. Then, we have three subcases to consider:
– two distinct real roots of multiplicity 2: The orbit induced at X2Y 2 is the image of the
PSL(2,R)-equivariant map
dS1,1 ⊂ P(R2[X,Y ]) −→ H2,2, [L] 7→ [L2],
where R2[X,Y ] is the vector space of homogeneous polynomials of degree 2 in two variables
X and Y , endowed with discriminant as a PSL(2,R)-invariant bilinear form of signature
(1, 2) [2, Section 5.3]. (Here, L is the projective class of a Lorentzian bilinear form on R2).
The vectors tangent to the orbit at X2Y 2 induced by the 1-parameter subgroups P and E
are vP = −2XY 3 and vE = 2X3Y − 2XY 3, respectively. It is easy to see that the 2-plane
generated by {vp, vE} is of signature (1, 1). Hence, the orbit induced at X2Y 2 is Lorentzian.
6 MASOUD HASSANI
– three distinct real roots, one of them being of multiplicity 2: Denote by M the orbit induced
by PSL(2,R) at [f ] = [XY 2(X − Y )]. The vectors tangent to M at [f ] induced by the
1-parameter subgroups H, P and E are:
vH = −2XY 3, vP = Y 4 − 2XY 3, vE = Y 4 −X4 − 2X2Y 2 +X3Y −XY 3,
respectively. Obviously, the lightlike vector vH + vP is orthogonal to T[f ]M . Therefore, the
restriction of the metric on T[f ]M is degenerate. It is easy to see that the quotient of T[f ]M
by the action of the isotropic line R(vH + vP ) is of signature (1, 1). Thus, M is of signature
(1, 1, 1).
– one real root of multiplicity 2, and one root in H2: Denote by M the orbit induced by
PSL(2,R) at [f ] = [Y 2(X2 + Y 2)]. The vectors tangent to M at [f ] induced by the 1-
parameter subgroups H, P and E are
vH = 4Y
4, vP = −2XY 3, vE = 2X3Y + 2XY 3,
respectively. Obviously, the lightlike vector vH is orthogonal T[f ]M . Therefore, the restriction
of the metric on T[f ]M is degenerate. It is easy to see that the quotient of T[f ]M by the
action of the isotropic line R(vH) is of signature (1, 1). Thus M is of signature (1, 1, 1).
• Two distinct real roots, and a complex root z in H2 making an angle θ smaller than 5π/6 with
the two real roots. Denote by M the orbit induced by PSL(2,R) at [f ] = [Y (X2 + Y 2)(X − rY )].
The vectors tangent to M at [f ] induced by the 1-parameter subgroups H, P and E are:
vH = −4rY 4 − 2X3Y + 2XY 3, vP = −3X2Y 2 + 2rXY 3 − Y 4,
vE = X
4 − Y 4 − 2rX3Y − 2rXY 3,
respectively. The following set of vectors is an orthogonal basis for T[f ]M where the first vector
is timelike and the two others are spacelike.
{(7r + 3r3)vH + (6− 2r2)vP + (5 + r2)vE , 4vP + vE , vH}.
Therefore, M is Lorentzian, i.e., of signature (1, 2). 
References
[1] T. Barbot, V. Charette, T. Drumm, W.M. Goldman, K. Melnick, A primer on the (2+1) Einstein universe. Recent
developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys, (2008) 179-229.
[2] B. Collier, N, Tholozan, J. Toulisse, The geometry of maximal representations of surface groups into SO(2, n).
https://arxiv.org/abs/1702.08799.
[3] A.J. Di Scala, T. Leistner, Connected subgroups of SO(2, n) acting irreducibly on R2,n. Inst. Hautes Études Sci. Publ.
Math. No. 78 (1993) 187-206.
[4] C. Frances, Géometrie et dynamique lorentziennes conformes. Thèse, E.N.S. Lyon (2002).
[5] M. Hassani, Cohomogeneity one actions on the three-dimensional Einstein universe. Work in Progress.
Université d’Avignon, Campus Jean-Henri Fabre- 301, rue Baruch de Spinoza, BP 21239 F-84 916 AVIGNON
Cedex 9.
E-mail address: masoud.hassani@alumni.univ-avignon.fr
University of Zanjan, Faculty of Science, Department of Mathematics, University blvd, Zanjan, Iran.
E-mail address: masoud.hasani@znu.ac.ir


ON THE IRREDUCIBLE ACTION OF PSL(2,R) ON THE 3-DIMENSIONAL EINSTEIN UNIVERSE

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On the irreducible action of psl(2, r) on the 3-dimensional einstein universe

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