hyperbolic dynamics - université paris-saclaylejan/pdfs/... · 2011. 2. 28. · iii.5) the...

Hyperbolic Dynamicsand

Brownian Motionsan introduction

Jacques FRANCHI and Yves LE JAN

January 2011

2

CONTENTS

Introduction page 5

Summary page 6

I) The Lorentz-Möbius group page 9

I.1) Lie algebras and groups : elementary introduction page 9

I.2) The Minkowski space and pseudo-metric page 18

I.3) The Lorentz-Möbius group and its Lie algebra page 20

I.4) Two remarkable subgroups of PSO(1, d) page 23

I.5) Structure of the elements of PSO(1, d) page 26

I.6) The hyperbolic space Hd and its boundary ∂Hd page 30I.7) Cartan and Iwasawa decompositions of PSO(1, d) page 32

II) Hyperbolic Geometry page 37

II.1) Geodesics and Light Rays page 37

II.2) A commutation relation page 46

II.3) Flows and leaves page 48

II.4) Structure of horospheres, and Busemann function page 52

III) Operators and Measures page 59

III.1) Casimir operator on PSO(1, d) page 59

III.2) Laplace operator D page 61III.3) Haar measure of PSO(1, d) page 64

III.4) The spherical Laplacian ∆Sd−1

page 73

III.5) The hyperbolic Laplacian ∆ page 76

III.6) Harmonic, Liouville and volume measures page 80

IV) Kleinian groups page 91

IV.1) Terminology page 91

IV.2) Dirichlet polyhedrons page 93

IV.3) Parabolic tesselation by an ideal 2n-gone page 94

IV.4) Examples of modular groups page 101

3

IV.5) 3-dimensional examples page 116

V) Measures and flows on Γ\Fd page 119V.1) Measures of Γ-invariant sets page 119

V.2) Ergodicity page 121

V.3) A mixing theorem page 124

V.4) Poincaré inequality page 129

VI) Basic Itô Calculus page 143

VI.1) Discrete martingales and stochastic integrals page 143

VI.2) Brownian Motion page 147

VI.3) Martingales in continuous time page 149

VI.4) The Itô integral page 154

VI.5) Itô’s Formula page 160

VI.6) Stratonovitch integral page 171

VII) Linear S. D. E.’s and B. M. on groups of matrices page 173

VII.1) Stochastic Differential Equations page 173

VII.2) Linear Stochastic Differential Equations page 178

VII.3) Approximation of left B.M. by exponentials page 192

VII.4) Lyapounov exponents page 199

VII.5) Diffusion processes page 201

VII.6) Examples of group-valued Brownian motions page 204

VIII) Central Limit Theorem for geodesics page 227

VIII.1) Adjoint Pd-valued left Brownian motions page 228VIII.2) Two dual diffusions page 234

VIII.3) Spectral gap along the foliation page 236

VIII.4) Resolvent kernel and conjugate functions page 242

VIII.5) Contour deformation page 245

VIII.6) Divergence of ωf page 249

VIII.7) Sinäı ’s Central Limit Theorem page 255

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IX) Appendix relating to geometry page 267

IX.1) Structure of symmetrical tensors in R1,d page 267IX.2) Another commutation relation in PSO(1, d) page 272

IX.3) The d’Alembertian 2 on R1,d page 277IX.5) Core-cusps decomposition page 282

X) Appendix relating to stochastic calculus page 287

X.1) A simple construction of real Brownian Motion page 287

X.2) Stochastic Riemanniann sums page 290

X.3) Chaos expansion page 292

XI) Index of notations, terms, and figures page 295

XI.1) General notations page 295

XI.2) Other notations page 296

XI.3) Index of Terms page 298

XI.4) Table of Figures page 304

References page 305

5

Introduction

This book provides first an elementary introduction to hyperbolic geometry, basedon the Lorentz group. Secondly, it introduces the hyperbolic Brownian motion andrelated diffusions on the Lorentz group. Thirdly, an analysis of the chaotic behaviourof the geodesic flow is performed using stochastic analysis methods. The main resultis Sinäı’s central limit theorem.

These methods had been exposed some years ago in research articles addressed toexperienced readers. In this book the necessary material of group theory and stochasticanalysis is exposed in a self-contained and voluntarily elementary way.

Only basic knowledge of linear algebra, calculus and probability theory is required.

Of course the reader familiar with hyperbolic geometry will traverse rapidly thefirst five chapters. Those who know stochastic analysis will do the same with the sixthchapter and the beginning of the seventh one.

Our approach of hyperbolic geometry is based on special relativity. The key role isplayed by the Lorentz-Möbius group PSO(1, d), Iwasawa decomposition, commutationrelations, Haar measure, and the hyperbolic Laplacian.

There is a lot of good expositions of stochastic analysis. We tried to make it as shortand elementary as possible, to the purpose of making it easily available to analysts andgeometers who could legitimately be reluctant to have to go through fifty pages beforegetting to the heart of the subject. Our exposition is closer to Itô’s and McKean’soriginal work (see [I], [MK]).

The main results and proofs (at least in the context of this book) are printed in largefont. The reader may at first glance through the remaining part, printed in smallerfont.

Finally, some related results, which are never used to prove the main results, butcomplete the expositions of stochastic calculus and hyperbolic geometry, are given inthe appendix. For the sake of completeness, the appendix also contains a constructionof Brownian motion.

6

Summary

I. The first chapter deals with the Lorentz group PSO(1, d), which is the (connectedcomponent of the unit in the) linear isometry group of Minkowski space-time. It isisomorphic to the Möbius group of direct hyperbolic isometries.

It begins with an elementary and short introduction to Lie algebras of matrices and as-sociated groups. Then the Minkowski space R1,d and its pseudo-metric are introduced,together with the Lorentz-Möbius group, and the space Fd of Lorentz frames, on whichPSO(1, d) acts both on the right and on the left.

We introduce then a subgroup Pd, generated by the first boost and the parabolictranslations, and we determine the conjugation classes of PSO(1, d).

The hyperbolic space Hd is defined as the unit pseudo-sphere of the Minkowski spaceR1,d. Iwasawa’s decomposition of PSO(1, d) is given, and yields Poincaré coordinatesin Hd. While the hyperbolic metric relates to Cartan’s decomposition of PSO(1, d).

II. The second chapter presents basic notions of hyperbolic geometry : geodesics,light rays, tangent bundle, etc. Then the geodesic and horocycle flows are defined, bya right action of Pd on frames. Horospheres and Busemann function are presented.

III. The third chapter deals with operators and measures. The Casimir operatorΞ on PSO(1, d) induces the Laplace operator D on Pd, and the hyperbolic Laplacian∆ . The Haar measure of PSO(1, d) is determined and shown to be bilateral. Thischapter ends with the presentation of harmonic, Liouville, and volume measures, andtheir analytical expressions.

IV. The fourth chapter deals with the geometric theory of Kleinian groups andtheir fundamental domains. It begins with the example of the parabolic tesselation ofthe hyperbolic plane by means of an ideal 2n-gone. Then Dirichlet polyhedrons andmodular groups are discussed, with Γ(2) and Γ(1) as main examples.

V. In the fifth chapter we consider measures of Γ-invariant sets, and establish amixing theorem. We derive a Poincaré inequality for the fundamental domain of ageneric geometrically finite, cofinite Kleinian group, i.e. a spectral gap for the corre-sponding Laplacian.

VI. The sixth chapter deals with the basic Itô calculus (Itô integral and for-mula), starting with a short account of the necessary background about martingalesand Brownian motion.

VII. The seventh chapter construction of (left and right) Brownian motions ongroups of matrices, as solutions to linear stochastic differential equations. We establish

7

in particular that the solution of such an equation lives in the subgroup associated tothe Lie subalgebra generated by the coefficients of the equation. Then we concentrateon important examples : the Heisenberg group, PSL(2), Pd and the Poincaré groupPd+1. We also introduce basic stochastic analysis on matrices, and use it to define thespherical and hyperbolic Brownian motions by means of a projection.

VIII. In the eighth chapter we provide a proof of the Sinäı Central Limit Theo-rem, generalised to the case of a geometrically finite and cofinite Kleinian group. Thistheorem indicates that asymptotically geodesics behave chaotically, and yields a quan-titative expression of this phenomenon. The method we use is by establishing suchresult first for Brownian trajectories, which is easier because of their strong indepen-dence properties. Then we compare geodesics with Brownian trajectories, by meansof a change of contour. This requires in particular to consider diffusion paths on thestable foliation and to derive the existence of a key potential kernel from the spectralgap exhibited in Chapter V.

Chapter I

The Lorentz-Möbius group

A large part of this book is centred on a careful analysis of this crucialgroup, to which this first chapter is mainly devoted.

I.1 Lie algebras and groups : introduction

I.1.1 M(d) and Lie subalgebras of M(d)

We shall here consider only algebras and groups of matrices, that is,subalgebras of the basic Lie algebra M(d), the set of all d×d real squarematrices (for some integer d ≥ 2), and subgroups of the basic Lie groupGL(d), the set of all d× d real square invertible matrices, known as thegeneral linear group.

The real vector space (M(d),+, ·) is made into an algebra, called aLie algebra, by means of the Lie bracket

[M,M ′] := MM ′ −M ′M = −[M ′,M ] ,MM ′ being the usual product of the square matrices M and M ′. The Liebracket satisfies clearly the Jacobi identity : for any matrices M,M ′,M ′′

we have

[[M,M ′],M ′′] + [[M ′,M ′′],M ] + [[M ′′,M ],M ′] = 0 .

9

10 CHAPTER I. THE LORENTZ-MÖBIUS GROUP

The adjoint action of M(d) on itself is defined by :

ad(M)(M ′) := [M,M ′] , for any M,M ′ ∈M(d) .

The Jacobi identity can be written as follows : for any M,M ′ ∈ G ,

ad([M,M ′]) = [ad(M), ad(M ′)](

= ad(M)◦ ad(M ′)−ad(M ′)◦ ad(M)).

The Adjoint action of the linear group GL(d) on M(d) is by conju-gation : GL(d) 3 g 7→ Ad(g) is a morphism of groups, defined by :

Ad(g)(M) := gMg−1 , for any g ∈ GL(d) , M ∈M(d) .

A simple relation between the ad and Ad actions is as follows : for anyg ∈ GL(d),M ∈M(d), we have :

ad(Ad(g)(M)

)= Ad(g) ◦ ad(M) ◦ Ad(g)−1 . (I.1)

Indeed, for any M ′ ∈ G we have :

ad[Ad(g)(M)

](M ′) = g[M, g−1M ′g]g−1 = Ad(g) ◦ ad(M) ◦Ad(g)−1(M ′).

Using the exponential map (whose definition is recalled in the followingsection I.1.2), we have moreover the following.

Lemma I.1.1.1 The differential of Ad◦exp at the unit 1 is ad . And :

Ad(exp(A)) = exp[ad(A)] on M(d) , for any A ∈M(d) . (I.2)

Proof Consider the analytical map t 7→ Φ(t) := Ad(exp(tA)) from Rinto the space of endomorphisms on M(d). It satisfies for any real t :

d

dtΦ(t) =

dods

Ad(exp((s+ t)A)) =dods

Ad(exp(sA) exp(tA))

=dods

Φ(s) ◦ Φ(t) = ad(A) ◦ Φ(t).

I.1. LIE ALGEBRAS AND GROUPS : INTRODUCTION 11

Hence, Φ solves a linear differential equation, which has the uniquesolution :

Φ(t) = exp[t ad(A)] ◦ Φ(0) = exp[t ad(A)] . �

A Lie algebra G of matrices is a vector subspace of M(d) which isstable under the Lie bracket [·, ·].The adjoint action of G defines then a linear map M 7→ ad(M) from G into the vector spaceof derivations on G. Indeed, the Jacobi identity is equivalent to :

ad(M)([M ′,M ′′]) = [ad(M)(M ′),M ′′] + [M ′, ad(M)(M ′′)] , for any M,M ′,M ′′ ∈ G .

The Killing form K of a Lie algebra G is the bilinear form on Gdefined (by means of the trace within G) by :

K(M ′,M ′′) := TrG(ad(M′) ◦ ad(M ′′)) , for any M ′,M ′′ ∈ G . (I.3)

The adjoint action acts skew-symmetrically on it, meaning that :

K(ad(M)(M ′),M ′′) = −K(M ′, ad(M)(M ′′)) , for all M,M ′,M ′′ ∈ G .Indeed, this is equivalent to : K([M,M ′],M ′′) = K(M, [M ′,M ′′]) for any M,M ′,M ′′, orusing the Jacobi identity under its second formulation above, to :

TrG(ad(M′) ◦ ad(M) ◦ ad(M ′′)) = TrG(ad(M) ◦ ad(M ′′) ◦ ad(M ′)), which holds clearly.

I.1.2 The exponential map

The exponential map is defined by

exp(M) :=∑n∈N

Mn/n! , for any M ∈M(d) ,

and is C∞ from M(d) into M(d). The formula exp(M + M ′) =exp(M) exp(M ′) is correct for commuting matrices M,M ′, but does nothold in general.


Denote as usual by Tr(M) :=d∑j=1

Mjj the trace of any matrix M ∈M(d), and by det(M) its determinant. The formula

det(exp(M)) = eTr(M)

holds trivially for diagonalisable matrices M , and hence holds for allM ∈M(d) by density. It shows that the range of exp is included in thegeneral linear group GL(d).

We denote by 1 the unit matrix, unit element of GL(d).

Proposition I.1.2.1 The differential d expM of the exponential mapat any M ∈M(d) expresses as follows : for any B ∈M(d),

d expM(B) :=dodε

exp(M + εB) =

(∑k∈N

1

(k + 1)!ad(M)k(B)

)exp(M) .

The exponential map induces a diffeomorphism from a neighbourhood of0 in M(d) onto a neighbourhood of exp(0) = 1 in GL(d).

Proof Denoting henceforth bydodε

the derivative at 0 with respect to

ε , we have :

d expM(B) =dodε

exp(M + εB) =∑n∈N

Yn(n+ 1)!

=∑n∈N

Zn(n+ 1)!

,

with Yn :=n∑k=0

Mn−kBMk , and Zn :=n∑k=0

Ck+1n+1 ad(M)k(B)Mn−k .

Indeed, let us prove by induction that Yn = Zn for n ∈ N : we haveY0 = B = Z0 , and assuming Yn−1 = Zn−1 , we have :

Yn = MnB + Yn−1M = M

nB + Zn−1M


= MnB +n−1∑k=0

Ck+1n ad(M)k(B)Mn−k

= MnB + Zn − ad(M)n(B)−n−1∑k=0

Cknad(M)k(B)Mn−k

= Zn +MnB −

n∑k=0

Cknad(M)k(B)Mn−k

= Zn +MnB −

(ad(M) + [A 7→ AM ]

)n(B) = Zn ,

since ad(M) and the right multiplication by M commute.

Finally, the above expression with Zn entails the expression of thestatement :

d expM(B) =∑

0≤k≤n


group associated to a Lie subalgebra G the subgroup of GL(d) gener-ated by exp(G).

As any group, such group G acts on itself by its inner automorphisms :G 3 g 7→ Ad(g), the conjugation by g already defined in Section I.1.1.

Lemma I.1.3.1 For any g ∈ G (associated to G), Ad(g) maps G to G.

Proof For any fixed A,B ∈ G and any real t , set :ϕ(t) := exp(tA)B exp(−tA) ∈M(d).

This defines a C∞ map ϕ from R intoM(d), solving the order one lineardifferential equation : ϕ′ = Aϕ − ϕA , and such that ϕ(0) = B ∈ G,ϕ′(0) = [A,B] ∈ G . Hence the equation satisfied by ϕ can be solved inthe vector space G, showing by uniqueness that ϕ must be G-valued.By an immediate induction, this proves that for any A1, . . . , An, B ∈ Gwe have Ad

(exp(A1) . . . exp(An)

)(B) ∈ G , that is : Ad(g)(B) ∈ G for

any g ∈ G . �The Killing form is invariant under inner automorphisms : for any

g ∈ G and M,M ′ ∈ G , we have

K(Ad(g)(M),Ad(g)(M ′)

)= K(M,M ′) . (I.4)

Indeed, by definition (I.3) of K and by Formula (I.1), we have :

K(Ad(g)(M),Ad(g)(M ′)

)= Tr

(ad[Ad(g)(M)

]◦ ad

[Ad(g)(M ′)

])= Tr

(Ad(g) ◦ ad(M) ◦ ad(M ′) ◦ Ad(g)−1

)= Tr

(ad(M) ◦ ad(M ′)

)= K(M,M ′) .

The group G associated to a Lie subalgebra G is a Lie subgroup of GL(d) when itsatisfies the following condition : there exists a neighbourhood U of 0 in M(d) such thatexp(U) ∩G = exp(U ∩ G).


This is consistent with the usual definition of a Lie group, see for example the introducingchapter of [Kn], or ([Ho], Chapter II, Section 2). In particular, it holds that G = {M ∈M(d) | exp(tM) ∈ G for any real t }. Moreover, any Lie subgroup G is necessarily closedin GL(d) : indeed, fixing g ∈ G , a compact neighbourhood V of 0 in M(d) on which therestriction of the exponential map is one-to-one and such that exp(V) ∩ G = exp(V ∩ G),another neighbourhood U of 0 inM(d) such that exp(−U) exp(U) ⊂ exp(V), and a sequence(gn) ⊂ G ∩ g exp(U) which converges to g , we have g−10 gn ∈ exp(V) ∩ G, whence g−10 gn =exp(hn), with some hn ∈ V ∩ G ; selecting a sub-sequence, we may suppose that hn → h ∈V ∩ G, whence g = g0 exp(h) ∈ G .

On the contrary, a group G associated to a Lie subalgebra G does not need to be closedin GL(d), as shows the following example :

the one-parameter subgroup H of GL(4) defined by : H := {exp[tÃ] | t ∈ R}, whereÃ :=

(A 00 π A

), with A :=

(0 −11 0

), does not contain −1 ∈ H, as is easily seen

by considering a sequence tn ∈ 2N + 1 approaching π modulo 2π .

Observe that, by continuity of the product and of the inverse maps, the closure in GL(d)

of a group associated to a Lie subalgebra G remains a subgroup of GL(d), which we shall callthe closed group associated to the Lie subalgebra G . This is a Lie group, since it is known(see ([Ho], Chapter VIII)) that any closed subgroup of a Lie group is also a Lie group.

In a general and usual way, any left action of any group G on anyset S is extended to any function F on this set, by setting :

LgF (s) := F (gs) , for any (g, s) ∈ G× S .Similarly, any right action of G on S is extended to functions, bysetting :

RgF (s) := F (sg) , for any (g, s) ∈ G× S .

We shall say that a real function f on a group G associated toa Lie subalgebra G is differentiable, when its restrictions to all lines[t 7→ g exp(t A)] and [t 7→ exp(t A) g] are, for any g ∈ G , A ∈ G . Thenwe define the right Lie derivative LAf by :

LAf(g) :=dodεf [g exp(εA)] , for any g ∈ G . (I.5)


This defines a left-invariant vector field LA on G , which means that itcommutes with left translations on G :

LA ◦ Lg = Lg ◦ LA , for all g ∈ G and A ∈ G . (I.6)

We have also

LA ◦Rg = Rg ◦ LAd(g−1)(A) , for all g ∈ G and A ∈ G . (I.7)

Similarly, the left Lie derivative L′Af is defined by :

L′Af(g) :=dodεf [exp(εA) g] , (I.8)

and defines a right-invariant vector field L′A on G :

L′A ◦Rg = Rg ◦ L′A , for all g ∈ G and A ∈ G ,

such that

L′A ◦ Lg = Lg ◦ L′Ad(g)(A) , for all g ∈ G and A ∈ G .

We have furthermore the following formula :

LARgF (h) = L′ALhF (g) , for any A ∈ G , g, h ∈ G . (I.9)

The map A 7→ LA from G onto the Lie algebra of left-invariantvector fields on G is an isomorphism of Lie algebras. We have indeedthe formula :

[LA,LB] = L[A,B] , for any A,B ∈ G . (I.10)

I.1.4 Basic examples

1) The Lie algebra sl(d) of traceless matrices, and the Lie group SL(d)of matrices having determinant one.


2) The Lie algebra so(d) of antisymmetric matrices, and the Lie groupSO(d) of rotation matrices.

3) The Lie algebra so(1, d) :={A ∈ M(d + 1)

∣∣ tA = −JAJ}, whereJ ∈ M(d + 1) denotes the diagonal matrix having diagonal entries(1,−1, . . . ,−1) (in that order). Our main interest will be the Lorentz-Möbius Lie group PSO(1, d) made of the g ∈ SL(d + 1) such thattgJg = J and [〈x, x〉 > 0 , x0 > 0] ⇒ (gx)0 > 0 . In words, this isthe group of matrices preserving the quadratic form defined on R1+d by :〈x, x〉 := x20 − x21 − · · · − x2d , the upper sheet of hyperboloid Hd := {x ∈R1+d | 〈x, x〉 = 1, x0 > 0}, and the orientation. This is the connectedcomponent of the unit element in the group O(1, d) of all isomorphismswhich preserve the Lorentz quadratic form 〈x, x〉. See Section I.3 below.We have also so(1, d) =

{A ∈M(d+1)

∣∣ 〈Ax, x〉 = 0 for all x ∈ R1+d}.Proposition I.1.4.1 The groups PSL(2) := SL(2)/{±1} andPSO(1, 2) are isomorphic. The Lie algebras sl(2) and so(1, 2) areisomorphic.

Proof The three matrices Y0 :=

(0 −11 0

), Y1 :=

(0 11 0

), Y2 :=

(1 00 −1

)constitute

a basis of sl(2), and we have det(x0Y0 + x1Y1 + x2Y2) = x20 − x21 − x22 . Since for any

g ∈ SL(2) the linear map Ad(g), acting on sl(2), preserves the determinant, and since SL(2)is connected and the morphism g 7→ Ad(g) is continuous, we see that the map Ad(g) belongsto the group O(1, 2), and even to the connected component of its unit element. Hence themorphism g 7→ Ad(g) maps the Lie group SL(2) into PSO(1, 2).Therefore, by differentiating at the unit element, we see that the morphism A 7→ ad(A) mapsthe Lie algebra sl(2) into the Lie algebra so(1, 2). Moreover, if ad(A) = 0, then A commuteswith all M ∈ sl(2) and then vanishes. Hence A 7→ ad(A) is indeed an isomorphism of Liealgebras (by the Jacobi identity).

Hence Ad(SL(2)) is a neighbourhood of the unit element and a subgroup of the connectedLie group PSO(1, 2), so that it must be the whole PSO(1, 2). Finally, if Ad(g) = 1, theng ∈ SL(2) must commute with the elements of sl(2), and then be ±1. �

The elements of the group PSL(2) identify with homographies of the Poincaré half plane


R×R∗+ ≡{z = x+

√−1 y ∈ C

∣∣ y > 0}, by ±(a bc d

)←→

[z 7→ az+bcz+d

].

We could have considered matrices with complex entries as well,Md(C) instead of M(d) ≡ Md(R). In this context, considering thegroup SL2(C) of spin-matrices, we have the following continuation ofProposition I.1.4.1.

Proposition I.1.4.2 The groups PSL2(C) := SL2(C)/{±1} andPSO(1, 3) are isomorphic. The Lie algebras sl2(C) and so(1, 3) areisomorphic.

Proof We follow the very similar proof of Proposition I.1.4.1. Let us identify the Minkowskispace R1,3 (see Section I.2 below) with the subset H2 of hermitian matrices in M2(C), bymeans of the map ξ 7→

(ξ0 + ξ1 ξ2 +

√−1 ξ3

ξ2 −√−1 ξ3 ξ0 − ξ1

), the determinant giving the Lorentz

quadratic form. Define the action ϕ(g) of g ∈ SL2(C) on H2 by H 7→ g H tg . Themorphism ϕ maps the group SL2(C) into PSO(1, 3).Differentiating, we get a morphism dϕ(1) from the Lie algebra sl2(C) into the Lie algebraso(1, 3), which have both six dimensions. The kernel of this morphism is made of thosematrices A ∈ M2(C) which have null trace and which satisfy AH + H tA = 0 for anyH ∈ H2 ; A = 0 being the only solution, the two algebras are indeed isomorphic.

Hence the range ϕ(SL2(C)) is the whole PSO(1, 3). Finally, if g ∈ SL2(C) belongs to thekernel of ϕ , then g H tg = H for any H ∈ H2, which implies g = ±1 . �

I.2 The Minkowski space and pseudo-metric

Fix an integer d ≥ 2 , and consider the Minkowski space :R1,d :=

{ξ = (ξ0, . . . , ξd) ∈ R× Rd

},

endowed with the Minkowski pseudo-metric (Lorentz quadratic form) :

〈ξ, ξ〉 := ξ20 −d∑j=1

ξ2j .

I.2. THE MINKOWSKI SPACE AND PSEUDO-METRIC 19

We denote by (e0, e1, . . . , ed) the canonical basis of R1,d, and we orien-tate R1,d by taking this basis as direct. We have 〈ei, ej〉 = 1{i=j=0} −1{i=j 6=0}.

Note that {e0, e1}⊥ is the subspace generated by {e2, . . . , ed} ; weidentify it with Rd−1 ; and similarly, we identify with Rd the subspace{e0}⊥, generated by {e1, . . . , ed} :

e⊥0 ≡ Rd and {e0, e1}⊥ ≡ Rd−1.

Note that the opposite of the pseudo-metric induces obviously the Eu-clidian metric on Rd. As usual, we shall denote by Sd−1, Sd−2 the cor-responding unit Euclidian spheres.

A vector ξ ∈ R1,d is called lightlike (or isotropic) if 〈ξ, ξ〉 = 0 , timelikeif 〈ξ, ξ〉 > 0 , positive timelike or future-directed if 〈ξ, ξ〉 > 0 and ξ0 >0 , spacelike if 〈ξ, ξ〉 < 0 , and non-spacelike if 〈ξ, ξ〉 ≥ 0 .

The light cone of R1,d is the upper half-cone of lightlike vectors of theMinkowski space :

C :={ξ ∈ R1,d

∣∣∣ 〈ξ, ξ〉 = 0 , ξ0 > 0}.The solid light cone of R1,d is the convex hull of C , i.e. the upper

half of the solid cone of timelike vectors :

C :={ξ ∈ R1,d

∣∣∣ 〈ξ, ξ〉 ≥ 0 , ξ0 > 0}.Lemma I.2.1 (i) No plane of R1,d is included in the solid cone ofnon-spacelike vectors.

(ii) For any ξ, ξ′ ∈ C , we have 〈ξ, ξ′〉 ≥√〈ξ, ξ〉〈ξ′, ξ′〉 , with equality

if and only if ξ, ξ′ are collinear.


Proof Set |ξ| :=√ξ20 − 〈ξ, ξ〉 , for any ξ ∈ R1,d.

(i) For non-collinear ξ, ξ′ ∈ R1,d such that 〈ξ, ξ〉 ≥ 0 and 〈ξ′, ξ′〉 ≥ 0 , we must have ξ0 6= 0and then

〈ξ − (ξ′0/ξ0)ξ′, ξ − (ξ′0/ξ0)ξ′

〉= −

∣∣ξ − (ξ′0/ξ0)ξ′∣∣2 < 0 .(ii) For ξ, ξ′ ∈ C , we have on one hand ξ0 ξ′0 ≥ |ξ| |ξ′| ≥

d∑j=1

ξj ξ′j , whence 〈ξ, ξ′〉 ≥ 0 , and

on the other hand :

〈ξ, ξ′〉2 ≥(ξ0ξ′0 − |ξ||ξ′|

)2= 〈ξ, ξ〉〈ξ′, ξ′〉+

(ξ0|ξ′| − ξ′0|ξ|

)2 ≥ 〈ξ, ξ〉〈ξ′, ξ′〉 .Hence 〈ξ, ξ′〉 ≥

√〈ξ, ξ〉〈ξ′, ξ′〉 , and equality occurs if and only if (ξ1, . . . , ξd) and (ξ′1, . . . , ξ′d)

are collinear(of same sense, by the case of equality in Schwarz inequality |ξ| |ξ′| ≥

d∑j=1

ξj ξ′j

)and ξ0|ξ′|−ξ′0|ξ| = 0 . Then |ξ| = 0 holds if and only if |ξ′| = 0 holds, in which case ξ, ξ′ areindeed collinear. Finally, for |ξ| 6= 0 , we have (ξ′1, . . . , ξ′d) = λ(ξ1, . . . , ξd) for some positiveλ , and then ξ′0 = λ ξ0 too. �

Definition I.2.2 A direct basis β = (β0, β1, . . . , βd) of R1,d such thatthe first component of β0 is positive, and such that 〈βi, βj〉 = 1{i=j=0}−1{i=j 6=0} for 0 ≤ i, j ≤ d (pseudo-orthonormality), will be called hence-forth a Lorentz frame. We set π0(β) := β0 .

The set of all Lorentz frames of R1,d will be denoted henceforth by Fd.

I.3 The Lorentz-Möbius group and its Lie algebra

Let PSO(1, d) denote the connected component of the unit ma-trix, in the Lorentz group O(1, d) of Lorentzian matrices (identifiedwith linear mappings of R1,d which preserve the pseudo-metric 〈·, ·〉).The special Lorentz group SO(1, d) is the subgroup (of index two inO(1, d)) of Lorentzian matrices which preserve the orientation, and theLorentz-Möbius group PSO(1, d) is the subgroup (of index two in

SO(1, d)) of Lorentzian matrices which preserve the light cone C (andthe orientation).

I.3. THE LORENTZ-MÖBIUS GROUP AND ITS LIE ALGEBRA 21

The special orthogonal group SO(d) (i.e. the rotation group of Rd)is identified with the subgroup of elements fixing the base vector e0 .Similarly, we identify henceforth with SO(d−1) the subgroup of elementsfixing both base vectors e0 and e1 .

As seen in Section I.1.4, the Lie algebra of PSO(1, d) is

so(1, d) ={A ∈M(d+ 1)

∣∣ 〈Ax, x〉 = 0 for all x ∈ R1,d}.We shall identify systematically the endomorphisms of R1,d with their

matrices in the canonical basis (e0, e1, .., ed) of R1,d.

The matrices

Ej := 〈e0, ·〉 ej − 〈ej, ·〉 e0 , for 1 ≤ j ≤ d ,

belong to the Lie algebra so(1, d), and generate so-called boosts (orhyperbolic screws) :

etEj(ξ0, . . . , ξj, . . .) = (ξ0 ch t+ ξj sh t, . . . , ξ0 sh t+ ξj ch t, . . .) ,

for any ξ ∈ R1,d, t ∈ R , 1 ≤ j ≤ d . The matrices

Ekl := 〈ek, ·〉 el − 〈el, ·〉 ek , for 1 ≤ k, l ≤ d ,

belong to the Lie algebra so(d) ⊂ so(1, d), and generate the subgroupSO(d). In a displayed form, we have (with for example d = 4, j = 2,k = 1, l = 3) :

Ej =

0 0 1 0 00 0 0 0 01 0 0 0 00 0 0 0 00 0 0 0 0

, Ekl =

0 0 0 0 00 0 0 1 00 0 0 0 00 −1 0 0 00 0 0 0 0

.


Proposition I.3.1 The matrices {Ej , Ekl | 1 ≤ j ≤ d, 1 ≤ k < l ≤ d}constitute a pseudo-orthonormal basis of so(1, d), endowed with its Kil-ling form K. Precisely, they are pair-wise orthogonal, and K(Ej, Ej) =2(d − 1) = −K(Ek`, Ek`). In particular, so(1, d) is a semisimple Liealgebra : its Killing form is non-degenerate.

Proof It is clear from its definition that so(1, d) has d(d + 1)/2 dimensions, and that theEj , Ekl above are linearly independent (this is also an obvious consequence of the computa-tion of K below), and therefore constitute a basis of so(1, d). Recall from Section I.1.1 thatthe Killing form K is defined by : K(E,E′) := Tr(ad(E) ◦ ad(E′)).From the definitions above of Ej , Ekl , we have easily (for 1 ≤ i, j, k, ` ≤ d) :

ad(Ej)(Ei) = Eji , ad(Ej)(Ek`) = − ad(Ek`)(Ej) = δjkE` − δjÈk , (I.11)

ad(Ek`)(Eij) = δiÈkj − δjÈki − δikE`j + δjkEì . (I.12)We deduce that (for 1 ≤ i, j, j′, k, ` ≤ d) :

ad(Ej′) ◦ ad(Ej)(Ei) = δjj′Ei − δij′Ej , ad(Ej′) ◦ ad(Ej)(Ek`) = δjkEj′` + δjÈkj′ ,

so thatK(Ej , Ej′) = (d− 1) δjj′ +

∑1≤k

TWO REMARKABLE SUBGROUPS OF PSO(1, d) 23

K(Ek`, Ek′`′) = (δk′`δk`′ − δkk′δ``′)×(

2 +∑j>k′

1 +∑j>`′

1 +∑i


Definition I.4.1 (i) Set for any r ∈ R :

θr := exp[r E1] =

ch r sh r 0 0 .. 0sh r ch r 0 0 .. 00 0 1 0 .. 0.. .. .. .. .. ..0 0 0 0 .. 1

.(ii) For any u = (0, 0, u2, .., ud) ∈ {e0, e1}⊥ ≡ Rd−1, set |u|2 =−〈u, u〉 =

d∑j=2

u2j , and :

θ+u := exp

[ d∑j=2

uj Ẽj

]=

1 + |u|

2

2 −|u|22 u2 .. .. ud

|u|22 1−

|u|22 u2 .. .. ud

u2 −u2 1 0 .. 0.. .. .. .. .. ..

ud −ud 0 .. 0 1

.

{θr | r ∈ R} and Td−1 := {θ+u |u ∈ Rd−1} are Abelian subgroups ofPSO(1, d).

Note that the above displayed expression for θ+u follows at once fromthe following observation :[

d∑j=2

uj Ẽj

]2=

d∑j=2

u2j Ẽ2j , and then

[d∑j=2

uj Ẽj

]3= 0 .

Note also that θ+u+v = θ+u θ

+v , so that Td−1 is isomorphic to Rd−1.

Denote then by τ̂d the Lie subalgebra of so(1, d) generated by τd−1and E1 . Note that [E1, Ej] = E1,j and [E1, E1,j] = Ej , so that[E1, Ẽj] = Ẽj , and τ̂d has dimension d .

Denote by Pd the Lie subgroup of PSO(1, d) generated by thesematrices, which is also the Lie subgroup of PSO(1, d) associated with

TWO REMARKABLE SUBGROUPS OF PSO(1, d) 25

the Lie subalgebra τ̂d of so(1, d). Note that θ+u (e0 + e1) = (e0 + e1) for

any u ∈ Rd−1 and that θr(e0 + e1) = er(e0 + e1) for any r ∈ R. Thus allmatrices of Pd fix the particular half line R∗+(e0 + e1) of the light cone C.

Proposition I.4.2 (i) Any element of the Lie subgroup Pd can bewritten θ+x θt in a unique way, for some (t, x) ∈ R× Rd−1. Moreover,for any (t, x) ∈ R× Rd−1 we have :

θt θ+x = θ

+etx θt . (I.14)

For any z = (x, y) ∈ Rd−1 × R∗+ , set Tz ≡ Tx,y := θ+x θlog y .We have thus Pd = {Tz | z ∈ Rd−1 × R∗+}, and the product formula :

Tx,y Tx′,y′ = Tx+yx′, yy′ . (I.15)

(ii) For any % ∈ SO(d) and 1 ≤ j ≤ d , we have

%Ej %−1 = E%(ej) := −

d∑k=1

〈%(ej), ek〉Ek ,

and for any % ∈ SO(d− 1), we have% θ+x %

−1 = θ+%(x) , and % θt %−1 = θt .

Proof (i) The commutation formula (I.14) follows directly from theexpressions of matrices θ+x , θt displayed above. The existence of thedecomposition θ+u θt follows at once. The uniqueness is clear, since θt =θ+u implies obviously t = 0, u = 0 , looking again at the expressionsdisplayed above for these matrices. Formula (I.15) follows at once fromFormula (I.14).

(ii) By the commutation formula (I.11), we have : [A,Ej] = EAej =

−d∑

k=1

〈Aej, ek〉Ek for any A ∈ so(d) and 1 ≤ j ≤ d , whence by (I.2) :

Ad(exp(A))(Ej) = Eexp(A)(ej) . The remaining assertions are straight-forward. �


Remark I.4.3 Formula (I.15) shows that the subgroup Pd is isomorphic to a semi-directproduct of R and Rd−1, namely the classical group of translations and dilatations of Rd−1,or equivalently, to the group of d× d triangular matrices :{(

1 0x y 1d−1

)∈ GL(d)

∣∣∣∣x ∈ Rd−1(written as a column), y > 0, unit 1d−1 ∈ GL(d− 1)},associating

(1 0x 1d−1

)with θ+x and

(1 00 y 1d−1

)with θlog y .

I.5 Structure of the elements of PSO(1, d)

Recall that any % ∈ SO(d) is conjugate (in SO(d)) to an element whosenon-null entries are either 1’s on the diagonal, or forming diagonal planar

rotation blocks :

[cosϕj − sinϕjsinϕj cosϕj

]. This is of course well known.

We determine here the structure of elements of PSO(1, d), and classifythem.

Consider the three following subgroups of PSO(1, d) : SO(d) ,

and PSO(1, 1)× SO(d− 1) and T1 × SO(d− 2), i.e. ch r sh r 0sh r ch r 0

0 0 R

= θrR = Rθr ∣∣∣∣ r ∈ R , R ∈ SO(d− 1) ,

and

1 + u2

2 −u2

2 u 0u2

2 1− u2

2 u 0u −u 1 00 0 0 %

= θ+ue2 % = % θ+ue2∣∣∣∣∣u ∈ R , % ∈ SO(d− 2)

.

STRUCTURE OF THE ELEMENTS OF PSO(1, d) 27

Note that the eigenvalues of the element

(ch r sh rsh r ch r

)are e±r (asso-

ciated with lightlike eigenvectors), and that

1 + u22 −u22 uu22 1− u

2

2 uu −u 1

hasthe unique eigenvalue 1 (associated with the unique lightlike eigenrayR+(1, 1, 0)).

Theorem I.5.1 Elements of PSO(1, d) can be classified as follows :

- Those with a timelike eigenvector are conjugated to an element ofSO(d), and will be called rotations (or elliptic elements).

- Those without any timelike eigenvector and with a unique lightlikeeigenvector are conjugated to an element of PSO(1, 1) × SO(d − 1),and will be called boosts (or loxodromic elements).

- Those without any timelike eigenvector and with two lightlike eigenvec-tors are conjugated to an element of T1 × SO(d− 2), and will be calledparabolic elements.

In other words,

- rotations are the elements of PSO(1, d) which fix at least a point of Hd ;- parabolic isometries are the elements of PSO(1, d) which fix no point of Hd and a uniquepoint of ∂Hd ;- boosts are the elements of PSO(1, d) which fix no point of Hd, but two points of ∂Hd (andnot 3).

Particularising to d = 2 and using Proposition I.1.4.1, it is immediately seen that an elementof PSL(2) is a rotation, parabolic, a boost, according as the absolute value of its trace is< 2 , = 2 , > 2 , respectively.

It appears that all eigenvalues of elements of PSO(1, d) are real positive or have modulus 1.

Proof 1) Let us complexify R1,d in C1,d, endowed with the sesquilinear pseudo-norm :

〈ξ, ξ′〉 := ξ0ξ′0 −d∑j=1

ξjξ′j , and fix γ ∈ PSO(1, d), linearly extended into an isometry of


C1,d, which can be diagonalised. If γv = λv , then γv̄ = λ̄v̄ , 〈v, v〉 = |λ|2〈v, v〉 , and〈v, v̄〉 = λ2〈v, v̄〉 . Therefore we have :

either |λ| = 1 or 〈v, v〉 = 0 ,

and :either λ = ±1 or 〈v, v̄〉 = 0 .

As a consequence, if λ = e√−1 ϕ /∈ R is an eigenvalue, with associated eigenvector v , we

must have 〈v, v̄〉 = 0 . Therefore, any real eigenvector associated with λ must be lightlike.2) Suppose there exists an eigenvalue λ such that |λ| 6= 1 , and let v denote an

associated eigenvector. Then 〈v, v〉 = 〈v, v̄〉 = 〈v̄, v̄〉 = 0 , which by Lemma I.2.1 forces thetwo real vectors (v + v̄) and (v − v̄)

√−1 to be linearly dependent : there exist real s, t non

vanishing both, such that s (v − v̄) =√−1 t (v + v̄), meaning that w := (s −

√−1 t) v is

real. Thus w is clearly an isotropic eigenvector associated to the eigenvalue λ , which wecan choose to belong to the light cone C . Note that this implies also that λ must be real,and even positive, since γ ∈ PSO(1, d) has to preserve C .

3) According to 2) above, consider a real eigenvalue λ = er′ 6= 1 , associated with an

eigenvector w ∈ C . Since det γ = 1 , we must have another eigenvalue λ′ having modulus6= 1 , necessarily also real and associated with some eigenvector w′ ∈ C . By Lemma I.2.1,we must have 〈w,w′〉 6= 0 , whence λ′ = 〈v, γv′〉

/〈w,w′〉 = 〈γ−1v, v′〉

/〈v, v′〉 = e−r′ . This

implies also that w,w′ are not collinear. Moreover v0 := w + 〈w,w′〉−1w′ is in Hd , andv1 := w − 〈w,w′〉−1w′ is such that 〈v1, v′1〉 = −1 . We get then at once, for some ε = ±1 :γv0 = (ch r)v0 + (sh r)v1 and γv1 = (sh r)v0 + (ch r)v1 .

The restriction of γ to the spacelike subspace {v0, v1}⊥ must be a rotation in this subspace,and then be a conjugate of some % ∈ SO(d− 1).

Furthermore, the eigenvectors different from v0±v1 are the (spacelike) eigenvectors of therotation part (as any other eigenvector could be decomposed into the sum of an eigenvectorbelonging to the timelike plane {v0, v1} and of an eigenvector in {v0, v1}⊥ with the sameeigenvalue). In particular, γ has no timelike eigenvector.

4) Suppose there is a timelike eigenvector v0 . By 1) and 3) above, the correspondingeigenvalue λ has to be ±1 , and actually 1 , since the solid light cone is preserved by γ .The restriction of γ to the spacelike subspace {v0}⊥ must be a rotation in this subspace,and then be a conjugate of some % ∈ SO(d), so that γ is a rotation.

5) We consider now the remaining possibility : eigenvalues have modulus 1, and there isno timelike eigenvector.

Consider an eigenvalue λ = e√−1 ϕ /∈ R, with associated eigenvector v . By 1) above we

must have 〈v, v̄〉 = 0 . Consider also the real vectors u := (v + v̄) and u′ := (v − v̄)√−1 .

STRUCTURE OF THE ELEMENTS OF PSO(1, d) 29

We have 〈u, u〉 = 〈v, v〉+ 〈v, v〉 = 〈u′, u′〉, and

γu = (cosϕ)u+ (sinϕ)u′ and γu′ = (cosϕ)u′ − (sinϕ)u , (∗)

whence 〈u, u′〉 = 〈γu, γu′〉 = cos(2ϕ) 〈u, u′〉 . Since cos(2ϕ) 6= 1 , this implies 0 = 〈u, u′〉,and then for all real s, t : 〈su + tu′, su + tu′〉 = (s2 + t2) 〈u, u〉 . Hence by Lemma I.2.1, uand u′ are either spacelike, or collinear and lightlike.

Now, if u 6= 0 and u′ = αu , using the above expressions (∗) for γu, γu′, we would haveα2 = −1 , a contradiction. Hence, u and u′ span a spacelike plane, and the restriction of γto this plane is a rotation. And the restriction of γ to {u, u′}⊥ is an element of PSO(1, d−2)which is as well neither a boost nor a rotation. Hence, by recursion, we are left with thecase where γ ∈ PSO(1, d − 2k) can only have ±1 as eigenvalues. Any such eigenvalue hasan associated real eigenvector, which must be non-timelike.

We can then restrict γ to the orthogonal of the space spanned by spacelike eigenvectors.

We are left with the case where all eigenvectors are lightlike. Let us show now that −1 cannotbe an eigenvalue : in that case, there would be u ∈ C such that γu = −u ; choosing someu′ ∈ C non-collinear to u , we should have 〈u, u′〉 > 0 , and 〈u, γu′〉 = 〈γ−1u, u′〉 < 0 , acontradiction since γu′ ∈ C .

It follows that the eigenvector is unique, since the sum of two non-collinear eigenvectorsin C would be timelike.

Note at this stage that the multiplicity of −1 in the decomposition of γ must be even,so that the restriction of γ to the subspace spanned by spacelike eigenvectors is a rotation.

6) We are finally left with γ ∈ PSO(1, d′) (with d′ = d − 2m > 0) possessing a uniqueeigenvector v ∈ C (up to a scalar), which is associated with the eigenvalue 1. We concludethe proof by showing that d′ = 2 , and that γ is conjugate to an element of T1 .Choose some v′ ∈ C such that 〈v, v′〉 = 12 , and set v′′ := γv′, v0 := v + v′, v1 := v − v′,u :=

√2〈v′, v′′〉 . Note that v′′ ∈ C , 〈v, v′′〉 = 12 , 〈v1, v1〉 = −〈v0, v0〉 = −1 , 〈v0, v1〉 = 0 , and

consider v2 := (v′′ − u2v − v′)/u . We have 〈v2, v〉 = 〈v2, v′〉 = 0 , 〈v2, v2〉 = −1 , so that

we can complete (v0, v1, v2) into a Lorentz basis (v0, v1, . . . , vd′). Let γ̃ denote the elementof PSO(1, d′) which has matrix θ+ue2 in this basis. we have then

γv′ = v′′ = u2v + v′ + u v2 = γ̃v′ and γv = v = γ̃v , whence γ̃−1γv0 = v0 and γ̃

−1γv1 = v1 .

Hence P := γ̃−1γ must belong to SO(d′− 1), i.e. have in the basis (v0, v1, . . . , vd′) a matrix

of the form

1 0 00 1 00 0 Q

, with Q ∈ SO(d′ − 1). We must also have for any λ :(λ−1)d′+1 = det(λ1−γ) = det(λ θ+−ue2−P ). Now, this last determinant is easily computed,by adding the first column to the second one, and then subtracting the second line to thefirst one. We get so : det(λ θ+−ue2 −P ) = (λ−1)2 det(λ1−Q), which entails det(λ1−Q) =


(λ − 1)d′−1, whence Q = 1 and then γ = γ̃ . Finally, since γ cannot have any spacelikeeigenvector, this forces d′ = 2 , and γ̃ is conjugate to θ+ue2 ∈ T1 . �

Remark I.5.2 Non-trivial boosts form a dense open set. The complement of this set isnegligible for the Haar measure of PSO(1, d) (defined by Formula (III.7) in Section III.3).

Proof By Theorem I.5.1, rotations and parabolic elements have all the eigenvalue 1, henceare included in the algebraic hypersurface having equation det(Tx,y %−1) = 0 (in the Iwasawacoordinates). The statement is now clear, by (III.7) and Theorem III.3.5. �

Remark I.5.3 The Lorentz-Möbius group is the image of its Lie algebra under the expo-nential map : PSO(1, d) = exp[so(1, d)].

Proof The property for γ ∈ PSO(1, d) to belong to the range of the exponential map isclearly stable under conjugation, and holds obviously true for SO(d), PSO(1, 1), and T1 , byDefinition I.4.1. �

I.6 The hyperbolic space Hd and its boundary ∂Hd

The set of vectors having pseudo-norm 1 and positive first coordi-nate is of particular interest, and constitutes the basic model for thehyperbolic space, which appears thus most naturally in the frameworkof Minkowski space.

Notation Let us denote by Hd the d-dimensional hyperbolic space,defined as the positive half of the unit pseudo-sphere of R1,d, that is tosay the hypersurface of R1,d made of all vectors having pseudo-norm 1and positive first coordinate :

Hd :={ξ = (ξ0, . . . , ξd) ∈ R1,d

∣∣ 〈ξ, ξ〉 = 1, ξ0 > 0}={

(ch r)e0 + (sh r)u∣∣ r ∈ R+, u ∈ Sd−1}.

This is of course a sheet of hyperboloid. Observe that any γ ∈ PSO(1, d)maps Hd onto Hd , as is clear by noticing that the elements ξ ∈ Hd are,

THE HYPERBOLIC SPACE Hd AND ITS BOUNDARY ∂Hd 31

among the vectors having pseudo-norm 1, those which satisfy 〈ξ, v〉 > 0 ,for any v ∈ C .

Proposition I.6.1 The Lie subgroup Pd acts transitively and properlyon Hd : for any p ∈ Hd , there exists a unique Tx,y ∈ Pd such thatTx,y e0 = p .

x, y are called the Poincaré coordinates of p .

In other words, we have in the canonical base e = (e0, . . . , ed), for aunique (x, y) ∈ Rd−1 × R∗+ :

p =(y2 + |x|2 + 1

2y

)e0 +

(y2 + |x|2 − 12y

)e1 +

d∑j=2

xj

yej . (I.16)

Equivalently, the Poincaré coordinates (x, y) of the point p ∈ Hd havingcoordinates (p0, . . . , pd) in the canonical base (e0, . . . , ed) are given by :

y =1

p0 − p1 =1

〈p, e0 + e1〉, and xj =

pj

p0 − p1 =−〈p, ej〉〈p, e0 + e1〉

(I.17)

for 2 ≤ j ≤ d .

The Poincaré coordinates (x, y) parametrize the subgroup Pd byRd−1 × R∗+ , which is usually called Poincaré (upper) half-space.Proof Formula (I.16) is merely given by the first column of the matrixTx,y (recall Definition I.4.1 and Proposition I.4.2). It is easily solved ina unique way, noticing that we must have |x|2 = y2

((p0)2 − (p1)2 − 1

),

which yields Formula (I.17). �

A light ray is a future-oriented lightlike direction, i.e. an element of :

∂Hd := C/R+ ={R+ ξ

∣∣ ξ ∈ C} .


In the projective space of R1,d, the set of light rays ∂Hd identifies withthe boundary of Hd. Note that the Lorentz-Möbius group PSO(1, d)acts on ∂Hd. Note also that for any light ray η ∈ ∂Hd, η⊥ is thehyperplane tangent to the light cone C at η (η⊥ contains η).

Note that the pseudo-metric of R1,d induces canonically a Euclidianstructure on each d-space p⊥, for any p ∈ Hd ; the inner product thereonbeing simply the opposite −〈·, ·〉 of the restriction of the pseudo-metric.

I.7 Cartan and Iwasawa decompositions of PSO(1, d)

We have the following Cartan decomposition of the Lorentz-Möbiusgroup PSO(1, d).

Theorem I.7.1 (Cartan decomposition of PSO(1, d)) Any γ in

PSO(1, d) can be written : γ = % θr %′ , with r ∈ R+ and % , %′ ∈ SO(d).

Moreover, we have :

(i) r = log ‖γ‖ = log ‖γ−1‖, where ‖γ‖ := max{|γ v|

/v ∈ Sd

}denotes

the operator norm of γ acting on Euclidian Rd+1, identified (as a vectorspace) with R1,d ;(ii) γ = %1 θr %

′1 holds if and only if %1 = % %̃ and %

′ = %̃ %′1 , with%̃ ∈ SO(d− 1) if r > 0.

Proof We must obviously have : γ e0 = % θr e0 ∈ Hd . For somer ∈ R+ and u ∈ Sd−1, we can write γ e0 = (ch r) e0 + (sh r)u. Denoteby % ∈ SO(d) the rotation acting trivially in {e1, u}⊥ and mapping e1to u : we have thus %−1γ e0 = (ch r) e0 +(sh r) e1 = θr e0 , meaning that%′ := θ−1r %

−1γ ∈ SO(d).(i) We have clearly ‖γ‖ = ‖% θr %′‖ = ‖θr‖, and, using the matrix

expression of θr , we get easily ‖θr‖ = er. Considering then the rotation

CARTAN AND IWASAWA DECOMPOSITIONS OF PSO(1, d) 33

%0 ∈ SO(d) fixing (e0, e3, . . . , ed) and mapping (e1, e2) to (−e1,−e2), wehave θ−r = %0 θr %0 , whence ‖γ−1‖ = ‖γ‖.

(ii) Suppose now that γ = % θr %′ = %1 θr %

′1 , and set %̃ := %

−1%1 .We have :

(ch r) e0 + (sh r) %̃ e1 = %̃ θr e0 = %̃ θr %′1 e0

= θr %′e0 = θr e0 = (ch r) e0 + (sh r) e1 ,

whence if r > 0 : %̃ e1 = e1 , meaning that %̃ ∈ SO(d−1). The remainingof (ii) is obvious, since θr commutes with any %̃ ∈ SO(d− 1). �Examples The Cartan decomposition of θt, θ

+u ∈ P2 is given by : for t < 0 and u ∈ R ,

θt = % θ|t| % , where % e1 + e1 = % e2 + e2 = 0 and % = 1 on {e1, e2}⊥ ; 1 + u2

2 −u2

2 uu2

2 1− u2

2 uu −u 1

= 1 0 00 u√u2+4 −2√u2+4

0 2√u2+4

u√u2+4

1 + u

2

2u2

√u2+4 0

u2

2

√u2+4 1 + u

2

2 00 0 1

1 0 00 −u√u2+4 2√u2+4

0 −2√u2+4

−u√u2+4

.We shall actually use mainly the following Iwasawa decomposition of

PSO(1, d), which asserts that PSO(1, d) = Pd SO(d), with uniqueness(recall Proposition I.4.2).

Theorem I.7.2 (Iwasawa decomposition of PSO(1, d))

Any γ ∈ PSO(1, d) can be written in a unique way :

γ = θ+u θt % , with θ+u θt ∈ Pd and % ∈ SO(d) .

Denote by Iw the canonical projection from PSO(1, d) onto Pd :

Iw(θ+u θt %) := θ+u θt .

Proof We must have : γ e0 = θ+u θt e0 ∈ Hd . By Proposition I.6.1,

this determines a unique θ+u θt ∈ Pd, proving the uniqueness. As tothe existence, fixing γ ∈ PSO(1, d) and using Proposition I.6.1 to get


θ+u θt ∈ Pd such that θ+u θt e0 = γ e0 , we have % := (θ+u θt)−1γ ∈ SO(d).�

The hyperbolic space Hd carries a natural metric, induced by R1,d,as follows.

Proposition I.7.3 Given two elements p, p′ of Hd, let β, β′ ∈ Fd betwo Lorentz frames such that β0 = p and β

′0 = p

′. Then the size r =log ‖β̃−1β̃′‖ =: dist (p, p′) of β̃−1β̃′ in its Cartan decomposition (recallTheorem I.7.1) depends only on (p, p′). It defines a metric on Hd, calledthe hyperbolic metric. Moreover, we have dist (p, p′) = argch [〈p, p′〉].

Proof According to Theorem I.7.1, write β̃−1β̃′ = % θr %′. Now, ifγ, γ′ ∈ Fd are such that γ0 = p and γ′0 = p′ too, then we have γ̃−1β̃ ∈SO(d) and γ̃′

−1β̃′ ∈ SO(d), so that γ̃−1γ̃′ =

(γ̃−1β̃ %

)θr(%′β̃′

−1γ̃′)

de-

termines the same r ∈ R+ as β̃−1β̃′. By Theorem I.7.1(i), we haver = log ‖β̃−1β̃′‖ , and then the triangle inequality is clear. And r = 0occurs if and only if β̃−1β̃′ ∈ SO(d), which means precisely that p = p′.Finally, we have :

argch[〈p, p′〉

]= argch

[〈β0, β′0〉

]= argch

[〈e0, β̃

−1β̃′(e0)〉R1,d]

= argch[〈e0, θr e0〉

]= r . �

It is obvious from Proposition I.7.3 that elements of PSO(1, d) define isometries of Hd.Actually all orientation-preserving hyperbolic isometries are got in this way.

Proposition I.7.4 The group of orientation-preserving hyperbolic isometries, id est oforientation-preserving isometries of the hyperbolic space Hd, is canonically isomorphic tothe Lorentz-Möbius group PSO(1, d).

Proof Consider first g, g′ ∈ PSO(1, d) which induce the same isometry of Hd. Theng′g−1 ∈ PSO(1, d) fixes all points of Hd, hence all points of the vector space generated byHd, meaning that this is the identity map.

CARTAN AND IWASAWA DECOMPOSITIONS OF PSO(1, d) 35

Consider then an orientation-preserving isometry f of Hd. Since PSO(1, d) acts transitivelyon Hd, we can suppose that f(e0) = e0 . This implies that 〈f(ξ), e0〉 = 〈ξ, e0〉 for any ξ ∈ Hd.The projection P := (ξ 7→ ξ − 〈ξ, e0〉 e0) is a bijection from Hd onto e⊥0 ≡ Rd. Considerf̃ := P ◦ f ◦ P−1 : Rd → Rd. For any v = P (ξ), v′ = P (ξ′) ∈ Rd, we have : 〈f̃(v), f̃(v′)〉 =〈f(ξ) − 〈f(ξ), e0〉 e0, f(ξ′) − 〈f(ξ′), e0〉 e0

〉= 〈f(ξ), f(ξ′)〉 − 〈f(ξ), e0〉〈f(ξ′), e0〉 = 〈ξ, ξ′〉 −

〈ξ, e0〉〈ξ′, e0〉 = 〈v, v′〉 . Thus f̃ ∈ SO(Rd), hence must be linear. We extend it by linearityto the whole R1,d, by setting f̃(e0) := e0 . We have thus an element f̃ ∈ SO(d) ⊂ PSO(1, d),which agrees with f on Hd, since for any ξ ∈ Hd :

f̃(ξ) = f̃(〈ξ, e0〉e0 + P (ξ)

)= 〈ξ, e0〉e0 + f̃(P (ξ)) = 〈f(ξ), e0〉e0 + P (f(ξ)) = f(ξ) .

Finally f ∈ PSO(1, d), since by linearity the preservation of orientation and of the pseudo-metric must extend from Hd to R1,d, and the preservation of Hd entails that of the lightcone. �

Proposition I.7.5 The hyperbolic distance between q1, q2 ∈ Hd, havingPoincaré coordinates (x1, y1), (x2, y2) respectively, expresses as :

dist (q1, q2) = argch[〈q1, q2〉

]= argch

[|x1 − x2|2 + y21 + y222 y1 y2

]. (I.18)

Proof We apply Propositions I.7.3 and I.6.1, and observe that by For-mula (I.15) we have :

〈q1, q2〉 = 〈Tx1,y1 e0 , Tx2,y2 e0〉 = 〈e0 , T−1x1,y1 Tx2,y2 e0〉 =〈e0 , Tx2−x1

y1,y2y1

e0

〉.

The result follows at once, using Formula (I.16). �

Remark I.7.6 The hyperbolic length of the line element, in the upper half-space Rd−1×R∗+of Poincaré coordinates, is the Euclidian one divided by the height y : ds =

√|dx|2+dy2

y.

This results indeed directly from Formula (I.18), since for small |δ|2 + ε2 we have :

dist (Tx,y e0 , Tx+δ,y+ε e0) = argch

[1 +

|δ|2 + ε22 y (y + ε)

]∼√|δ|2 + ε2y (y + ε)

∼√|δ|2 + ε2y

.

Chapter II

Hyperbolic Geometry

II.1 Geodesics and Light Rays

II.1.1 Hyperbolic geodesics

Definition II.1.1.1 Let us call geodesic of the hyperbolic space Hd anynon-empty intersection of Hd with a vector plane of R1,d. Thus, the setof geodesics of Hd identifies with the set of vector planes of R1,d whichintersect Hd.

The following remark justifies the above definition, and yields a naturalidentification between the set of geodesics of Hd and the set of pairs{η, η′} of distinct η, η′ ∈ ∂Hd, or between the set of oriented geodesicsof Hd and the set of ordered pairs (η, η′) of distinct η, η′ ∈ ∂Hd.Indeed, for η 6= η′ ∈ ∂Hd, and for non-null ξ ∈ η, ξ ∈ η′, we have 〈ξ, ξ′〉 > 0 by Lemma I.2.1,and then (2〈ξ, ξ′〉)−1(ξ + ξ′) ∈ Hd : the plane cone generated by η, η′ does intersect Hd.

Remark II.1.1.2 There exists a unique geodesic containing any twogiven distinct points of Hd ∪ ∂Hd. The geodesic containing the twopoints of ∂Hd fixed by a boost is called its axis.

37

38 CHAPTER II. HYPERBOLIC GEOMETRY

Proposition II.1.1.3 In two dimensions, consider two oriented distinct geodesics γ, γ′ ofH2, determined (up to orientation) by distinct planes P, P ′ respectively.- If P ∩ P ′ is timelike, then there exists a unique isometry ψ ∈ PSO(1, 2) mapping γ to γ′and such that ψ(P ∩ P ′) = P ∩ P ′, and ψ is a rotation.- If P ∩ P ′ is spacelike, then there exists a unique isometry ψ ∈ PSO(1, 2) mapping γ toγ′ and such that ψ(P ∩ P ′) = P ∩ P ′, and ψ is a boost. Moreover, there exists a uniquegeodesic γ′′ intersecting γ and γ′, and perpendicular to both of them, and γ′′ is the axis ofthe boost ψ .

- If P ∩P ′ is lightlike, then there exists a unique parabolic isometry ψ ∈ PSO(1, 2) mappingthe unoriented γ to the unoriented γ′ and such that ψ(P ∩ P ′) = P ∩ P ′.

There is no longer uniqueness in larger dimensions.

Proof If P ∩ P ′ is spacelike, its orthogonal is a plane intersecting H2 (it must containa timelike vector), hence a geodesic, clearly perpendicular to both P and P ′. Consider aLorentz frame β such that β2 ∈ P ∩ P ′ and β0 ∈ P . P ′ contains p ∈ H2 ∩ (P ∩ P ′)⊥,necessarily of the form : p = ch r β0 + sh r β1 with r ∈ R∗. Any solution ψ must map β0to p, and β2 to ±β2, whence a unique possibility, depending on the orientations of γ, γ′.Obviously, the boost having matrix θr in the frame β is the unique solution, and its axis is(P ∩ P ′)⊥, as claimed.

If P ∩ P ′ is timelike, any solution is a rotation according to Theorem I.5.1, which in anyLorentz frame β such that β0 ∈ P ∩ P ′ must have its matrix in SO(2). Obviously, a uniqueone meets the case, taking the orientations of γ, γ′ into account.

If P ∩P ′ is lightlike, consider a Lorentz frame β such that β0 ∈ P and β0 +β1 ∈ P ∩P ′.We can write the ends of γ, γ′ respectively

{R+(β0 + β1),R+(β0 − β1)

}and{

R+(β0 + β1),R+(β0 − cosϕβ1 + sinϕβ2)}

, with sinϕ 6= 0 . Then the parabolic isometry ψhaving matrix θ+tg (ϕ/2) in the frame β meets the case. If ψ

′ is another solution, then ψ−1◦ψ′

fixes R+(β0 +β1) and R+(β0−β1), and then (by Theorem I.5.1) is a boost, which must havematrix θr in the frame β. Thus ψ

′ has in the frame β matrix θ+tg (ϕ/2)θr , which happens to

have eigenvalues 1, er, (1 + tg 2(ϕ/2))e−r. By Theorem I.5.1, since ψ′ must be parabolic, this

forces r = 0 , hence ψ′ = ψ . �

II.1.2 Projection onto a light ray, and tangent bundle

We have the following useful projection, from the hyperbolic spaceonto a light ray.

II.1. GEODESICS AND LIGHT RAYS 39

Proposition II.1.2.1 For any (p, η) ∈ Hd×∂Hd, there exists a unique(pη, ηp) ∈ R1,d × η such that

〈p, pη〉 = 0 , 〈pη, pη〉 = −1 , and p+ pη = ηp (6= 0).

Proof Consider p′ := p − α η0 , for any given η0 ∈ η ∩ C . Then〈p, η0〉 > 0 , and 〈p, p′〉 = 0 ⇔ 〈p′, p′〉 = −1 ⇔ α 〈p, η0〉 = 1 showsthat there is indeed a unique solution. �Reciprocally, if q belongs to the unit sphere of p⊥, id est if 〈p, q〉 = 0and 〈q, q〉 = −1 , then η := R+(p+ q) ∈ ∂Hd, and pη = q .

Notations For any p ∈ Hd, denote by Jp the one-to-one map from∂Hd into p⊥ defined by : Jp(η) = pη . Its range is the unit sphere of theEuclidian d-space p⊥.

Let us denote by T 1Hd the unit tangent bundle of Hd, defined by :

T 1Hd :={

(p, q) ∈ Hd × R1,d∣∣∣ q ∈ p⊥, 〈q, q〉 = −1}.

Let us identify Hd × ∂Hd with the unit tangent bundle T 1Hd, bymeans of the bijection J , defined by : J(p, η) := (p, pη) = (p, Jp(η)).

Denote then by J ′ the map from Hd × ∂Hd into ∂Hd, which to any(p, η) associates the light ray η′ such that the intersection of the lightcone C with the plane generated by {p, η} be η ∪ η′. In other words :J ′(p, η) := R+(p− Jp(η)).

The projection π0 (recall Definition I.2.2) goes from Fd onto Hd. Letus define also a projection π1 from Fd onto Hd × ∂Hd, by : for anyβ ∈ Fd,

π1(β) :=(β0, R+(β0 + β1)

)=(π0(β), J

−1β0

(β1))∈ Hd × ∂Hd .


d

Figure II.1: projection from Hd onto a light ray

Note that by the above definitions, using the right action of PSO(1, d)on Fd (recall Remark I.3.2), we have at once the following identifications :

Fd/SO(d) ≡ π0(Fd) = Hd , and Fd/SO(d− 1) ≡ π1(Fd) = Hd× ∂Hd .

Proposition II.1.2.2 A sequence (qn) ⊂ Hd converges to η ∈ ∂Hd in the projective spaceof R1,d if and only if 〈qn, pη〉〈qn, p〉

goes to −1 for any p ∈ Hd, or equivalently, for one p ∈ Hd.This implies that dist (p, qn) goes to infinity.

Proof Take a Lorentz frame β ∈ Fd such that β0 = p and β1 = pη , and write qn =d∑j=0

qjn βj . Then convergence in the projective space of R1,d holds if and only if q0n → +∞

and qjn/q0n → 1{j=1} . Now, since |q0n|2 −d∑j=1|qjn|2 = 1 , this holds if and only if q1n/q0n → 1 .

As q0n = 〈qn, p〉 = ch [dist (p, qn)] and q1n = −〈qn, pη〉 , the proof is complete. �

Proposition II.1.2.3 Let {η′, η} be a geodesic of Hd, and let q ∈Hd \ {η′, η}. We have the following expression for the hyperbolic distancefrom the point q to the geodesic {η′, η}, valid for any point p on the


geodesic {η′, η} :

ch 2[dist (q, {η′, η})

]=

2

〈ηq, η′q〉= 〈q, ηp〉 × 〈q, η′p〉 . (II.1)

Moreover, the minimising geodesic from q to {η′, η} intersects the plane{η′, η} orthogonally.

Proof Given q ∈ Hd \ {η′, η}, it is straightforwardly verified that q′ := 〈ηq, η′q〉−1(ηq + η′q)is the pseudo-orthogonal projection of q on the vector plane generated by {ηq, η′q}, and thatq̄ := (2〈ηq, η′q〉)−1/2(ηq + η′q) belongs to the geodesic {η′, η}. Recall that we have 〈ηq, η′q〉 > 0by Lemma I.2.1. By Proposition I.7.3, we have ch 2[dist (q, q̄)] = 〈q, q̄〉2 = 2

/〈ηq, η′q〉 .

Let p be any point on the geodesic {η′, η}. This implies p = x ηq + x′ η′q, with x, x′ > 0and 2xx′〈ηq, η′q〉 = 1 . Hence, we have :

ch [dist (q, p)]− ch [dist (q, q̄)] = 〈p, q〉 − 〈q̄, q〉 = x+ x′ −√

2/〈ηq, η′q〉 =[√

x−√x′]2≥ 0 ,

which shows that dist (q, q̄) does realize dist[q, {η′, η}

].

Applying the preceding to p instead of q , we get 2 = 〈ηp, η′p〉 . As we have ηp = 〈q, ηp〉 ηqand η′p = 〈q, η′p〉 η′q , we find indeed 2 = 〈ηp, η′p〉 = 〈q, ηp〉〈q, η′p〉〈ηq, η′q〉.

Finally, qε :=[1 + 2ε

√2/〈ηq, η′q〉 + ε2

]−1/2(q̄ + ε q) runs the geodesic {q̄, q}, so that a

tangent vector at q̄ to this geodesic isdodεqε = q −

√2/〈ηq, η′q〉 q̄ = q − q′ , and the proof is

complete. �

We have then the following statement, similar to Proposition I.6.1,about the action of the subgroup Td−1 on the boundary ∂Hd, whichextends the Poincaré coordinates to the boundary ∂Hd. Recall thatθ+u (e0 + e1) = (e0 + e1), and then that Tx,y(e0 + e1) = y (e0 + e1).

Proposition II.1.2.4 The Lie subgroup Td−1 of horizontal transla-tions acts transitively and properly on ∂Hd \ R+(e0 + e1) : for any lightray η ∈ ∂Hd \ R+(e0 + e1), there exists a unique θ+u ∈ Td−1 such thatθ+u (e0 − e1) ∈ η . u is called the Poincaré coordinate of η .By convention, the Poincaré coordinate of R+(e0 + e1) will be ∞ .


In other words, we have in the canonical base e = (e0, . . . , ed), for aunique u ∈ Rd−1 :

ηe0 = e0 +

( |u|2 − 1|u|2 + 1

)e1 +

2

|u|2 + 1d∑j=2

uj ej . (II.2)

Equivalently, the Poincaré coordinate u of the light ray η in

∂Hd \ R+(e0 + e1) having coordinates proportional to (η0, . . . , ηd) in thecanonical base (e0, . . . , ed) is given by :

uj =ηj

η0 − η1 =−〈ηe0, ej〉〈ηe0, e0 + e1〉

for 2 ≤ j ≤ d . (II.3)

Proof Formula (II.2) is merely given by subtracting the two firstcolumns of the matrix θ+u (recall Definition I.4.1) and using PropositionII.1.2.1. It is then easily solved in a unique way, which yields Formula(II.3). �

Remark II.1.2.5 Consider a sequence (qn) ⊂ Hd, having Poincaré coordinates (xn, yn),and a light ray η ∈ ∂Hd, having Poincaré coordinate u (as in Propositions I.6.1 and II.1.2.4).Then the sequence (qn) goes to the boundary point η (recall Proposition II.1.2.2) if and onlyif its Poincaré coordinates (xn, yn) ∈ Rd−1 × R∗+ go to (u, 0) in the Euclidian topology ofRd−1 ×R∗+ . Check this, as an exercise.

II.1.3 Harmonic conjugation

Proposition II.1.3.1 Consider two pairs {η, η′} and {η′′, η′′′} of distinct light rays in ∂Hd.Then the three following statements are equivalent.

(i) The intersection of the two vector planes defined by {η, η′} and {η′′, η′′′} is a line, andthese planes are perpendicular.

(ii) The two geodesics defined by the vector planes {η, η′} and {η′′, η′′′} intersect orthogonallyin Hd.

(iii)〈η, η′〉〈η′′, η′′′〉〈η, η′′′〉〈η′, η′′〉 = 4 and

〈η, η′′〉〈η′, η′′′〉〈η, η′′′〉〈η′, η′′〉 = 1 . (Note that these fractions make sense,

since by homogeneity we can think for each of these light rays of any spanning vector.)


Moreover, if these conditions are fulfilled, then

(iv) the intersection of the geodesics {η, η′} and {η′′, η′′′} is the point of Hd which belongs to√〈η, η′′〉〈η, η′′′〉 η′ +

√〈η′, η′′〉〈η′, η′′′〉 η (note that this is a well defined timelike direction ;

in particular it does not depend on the vectors we can choose in C to represent the light raysη, η′, η′′, η′′′) ;

(v) there exists g ∈ PSO(1, d) which exchanges at the same time η and η′, and η′′ and η′′′:g(η) = η′, g(η′) = η, g(η′′) = η′′′, g(η′′′) = η′′.

Remark II.1.3.2 (i) The conditions of Item (iii) in Proposition II.1.3.1 are not redundantfor d ≥ 3, as show the two following examples. Taking (in the canonical frame of R1,3) :η = R+(1, 1, 0, 0), η′ = R+(1,−1, 0, 0), η′′ = R+(1, 0, 1, 0), and- either η′′′ = R+(1, 0, 0, 1), we get 2 for the first cross-ratio and 1 for the second ;- or η′′′ = (2, 1, 0,

√3 ), we get 4 for the first cross-ratio and 3 for the second ;

and in both cases, the intersection of both planes reduces to {0}.(ii) However, if d = 2, then the second condition entails the first one, while the recip-

rocal does not hold. Indeed, we can use a Lorentz frame such that η = R+(1, 1, 0), η′ =R+(1,−1, 0), and for η′′ = R+(1, cosα, sinα), η′′′ = R+(1, cosϕ, sinϕ), the second cross-ratio equals 1 if and only if tg 2(α/2) = tg 2(ϕ/2), i.e. if and only if α+ ϕ ∈ 2πZ, while thefirst cross-ratio equals 4 if and only if [tg (α/2)−3 tg (ϕ/2)][tg (α/2)+tg (ϕ/2)]cotg (ϕ/2) = 0 .

(iii) Note that by definition of the pseudo-norm on the exterior algebra :

〈ηp ∧ η′p, η′′p ∧ η′′′p 〉 = det(〈ηp, η′′p〉〈ηp, η′′′p 〉〈η′p, η′′p〉〈η′p, η′′′p 〉

)= 〈ηp, η′′p〉〈η′p, η′′′p 〉 − 〈ηp, η′′′p 〉〈η′p, η′′p〉 ,

that we have :〈ηp, η′′p〉〈η′p, η′′′p 〉〈ηp, η′′′p 〉〈η′p, η′′p〉

= 1 ⇔ 〈ηp ∧ η′p, η′′p ∧ η′′′p 〉 = 0 .

Definition II.1.3.3 Two pairs of distinct light rays {η, η′} and {η′′, η′′′} which satisfy theconditions of Proposition II.1.3.1, will be called harmonically conjugate.

In this case, the ideal quadrangle {η, η′′, η′, η′′′} will be called a harmonic quadrangle.

Proof of Proposition II.1.3.1. We use Proposition II.1.2.1.

Suppose first that condition (i) holds, pick a non-null vector q in the intersection of thevector planes {η, η′} and {η′′, η′′′}, and pick also some reference point p ∈ Hd. We haveq = αηp + βη

′p = γη

′′p + δη

′′′p , and there exists some non-null vector aηp + bη

′p orthogonal to

η′′p , η′′′p . This implies 0 = 〈αηp +βη′p, aηp + bη′p〉 = (αb+aβ) 〈ηp, η′p〉 , and then αb+aβ = 0 ,

so that we can suppose a = α , b = −β . Then we have 0 = 〈αηp − βη′p, η′′p〉 or equivalently


α〈ηp, η′′p〉 = β〈η′p, η′′p〉, which implies αβ > 0 . Now, since 〈q, q〉 = 2αβ 〈ηp, η′p〉 > 0 , we cansuppose that q ∈ Hd (up to multiplying it by a scalar), so that it must belong to bothgeodesics of Hd defined by {η, η′} and {η′′, η′′′}, and we can then take p = q .The tangent at p to the line {η, η′}∩Hd is limit of chords joining p to (1 + 2ε)−1/2(p+ εηp),so that lim

ε→0ε−1[(1 + 2ε)−1/2(p+ εηp)− p] = ηp − p = pη spans this tangent. Since it is also

orthogonal to p , it must be collinear to αηp − βη′p , and then orthogonal to η′′p , η′′′p .Thus (i)⇒ (ii) is proved.

Suppose then that condition (ii) holds, and denote by p the intersection of the geodesiclines defined by {η, η′} and {η′′, η′′′}. As we just saw in the proof of (i) ⇒ (ii) above, thenon-null vector pη is tangent to the line {η, η′}∩Hd and orthogonal to p , and similarly pη′′is tangent to the line {η′′, η′′′′} ∩Hd. Hence pη belongs the plane {η, η′} and is orthogonalto {p, pη′′}, hence to η′′, η′′′ . This proves (ii)⇒ (i), whence (i)⇔ (ii).

Moreover (still under hypothesis (ii)) p = αηp + βη′p = (α + β)p + (αpη + βpη′) implies

α + β = 1 and αpη + βpη′ = 0 , and then α = β =12 and pη′ = −pη . And similarly

pη′′′ = −pη′′ . Whence ηp + η′p = 2p , and 〈ηp, η′p〉 = 〈η′′p , η′′′p 〉 = 2 .We must also have 〈pη, pη′′〉 = 0 . Whence 〈ηp, η′′p〉 = 〈ηp, η′′′p 〉 = 〈η′p, η′′p〉 = 〈η′p, η′′′p 〉 = 1 .These values obviously satisfy condition (iii), so that (ii)⇒ (iii) and (iv) is proved.

Furthermore, we can complete (p, pη, pη′′) into some Lorentz frame (p, pη, pη′′ , p3, . . . , pd),and consider the isomorphism g which fixes p, p3, . . . , pd and maps (pη, pη′′) on

(−pη,−pη′′) : it belongs to PSO(1, d) and is as in (v). This proves (ii)⇒ (v).

Suppose reciprocally that condition (iii) holds, fix some reference point p ∈ Hd, andconsider q0 :=

√〈ηp, η′′p〉〈ηp, η′′′p 〉 η′p +

√〈η′p, η′′p〉〈η′p, η′′′p 〉 ηp , and

q :=q0√〈q0, q0〉

= (2 〈ηp, η′p〉)−12

[(〈ηp, η′′p〉〈ηp, η′′′p 〉〈η′p, η′′p〉〈η′p, η′′′p 〉

) 14

η′p +

(〈η′p, η′′p〉〈η′p, η′′′p 〉〈ηp, η′′p〉〈ηp, η′′′p 〉

) 14

ηp

]∈ Hd.

Similarly, set

q′ := (2 〈η′′p , η′′′p 〉)−12

[( 〈ηp, η′′p〉〈η′p, η′′p〉〈ηp, η′′′p 〉〈η′p, η′′′p 〉

) 14

η′′′p +

(〈ηp, η′′′p 〉〈η′p, η′′′p 〉〈ηp, η′′p〉〈η′p, η′′p〉

) 14

η′′p

]∈ Hd.

We have then : 〈q, q′〉 =√〈ηp, η′′p〉〈η′p, η′′′p 〉〈ηp, η′p〉〈η′′p , η′′′p 〉

+

√〈ηp, η′′′p 〉〈η′p, η′′p〉〈ηp, η′p〉〈η′′p , η′′′p 〉

= 1 by (iii).

Now, this means that q = q′ , so that q0 is indeed a non-null vector belonging to both planes{η, η′} and {η′′, η′′′}.


Consider then u :=√〈η′p, η′′p〉〈η′p, η′′′p 〉 ηp −

√〈ηp, η′′p〉〈ηp, η′′′p 〉 η′p , which is clearly orthogo-

nal to q . By (iii) we have :

〈u, η′′p〉 =√〈η′p, η′′p〉〈η′p, η′′′p 〉〈ηp, η′′p〉 −

√〈ηp, η′′p〉〈ηp, η′′′p 〉〈η′p, η′′p〉 = 0 ,

which means that the non-null vector u of the plane {η, η′} is orthogonal to the plane{η′′, η′′′}. This proves that (iii)⇒ (i), thereby concluding the proof. �

Proposition II.1.3.4 Consider two pairs {η, η′} and {η′′, η′′′} of distinct light rays in ∂Hd,and their Poincaré coordinates (recall Proposition II.1.2.4) u, u′, u′′, u′′′ respectively.

Then {η, η′} and {η′′, η′′′} are harmonically conjugate if and only if|u− u′| × |u′′ − u′′′||u− u′′′| × |u′ − u′′| = 2 and

|u− u′′| × |u′ − u′′′||u− u′′′| × |u′ − u′′| = 1 .

If d = 2 (and then u, u′, u′′, u′′′ ∈ R ∪ {∞}), this is equivalent to the more usual cross-ratiocondition :

[u, u′, u′′, u′′′] :=u′′ − uu′′ − u′ ×

u′′′ − u′u′′′ − u = −1 .

Proof The first claim is merely the transcription of Condition (iii) of Proposition II.1.3.1in terms of the Poincaré coordinates, since by Definition I.4.1(ii) we have simply〈

θ+u1(e0 − e1), θ+u2(e0 − e1)〉

=〈θ+u1−u2(e0 − e1), (e0 − e1)

〉= 2 |u1 − u2|2.

For d = 2 , this yields ε, ε′ ∈ {±1} such that[u, u′, u′′, u′′′] = ε and (u

′−u)×(u′′′−u′′)(u′′−u′)×(u′′′−u) = 2ε

′ .

But writing (u′−u) = (u′−u′′′) + (u′′′−u) and (u′′′−u′′) = (u′′′−u′) + (u′−u′′), we get atonce

(u′ − u)× (u′′′ − u′′)(u′′ − u′)× (u′′′ − u) = [u, u

′, u′′, u′′′]− 1 , so that the condition reduces to 2ε′ = ε−1 ,which is clearly equivalent to ε = ε′ = −1 , or to ε = −1 as well. �

Remark II.1.3.5 The Lorentz-Möbius group PSO(1, 2) acts transitively on the set ofharmonic quadrangles of H2. Any harmonic quadrangle is isometric to the quadrangle{−1, 0, 1,∞} of the Poincaré half-plane R×R∗+ .

Proof Of course, any isometry maps plainly a harmonic quadrangle on a harmonic quad-rangle. Reciprocally, by Remark II.3.4, a change of Lorentz frame, hence an isometry, mapsa given harmonic quadrangle onto another given one.

Let us however give an alternative proof that all harmonic quadrangles are isometric, con-

sidering the half-plane R × R∗+ and identifying an ideal vertex (i.e. a boundary point, or


light ray) with its Poincaré coordinate by means of Proposition II.1.2.4. By using a first

homography (seen as an element of SL(2), recall Proposition I.1.4.1), we move one vertex to

∞. We move next the most left vertex (on the real line) to −1 by a horizontal translation,and then the right neighbouring vertex of −1 to 0 by a dilatation (centered at −1). So far,we have got the new quadrangle {−1, 0, α,∞}, harmonic too, so that the geodesics [−1, α]and [0,∞] are orthogonal, which forces finally α = 1 : any harmonic quadrangle is indeedisometric to {−1, 0, 1,∞}. �

II.2 A commutation relation

We establish here the part we shall need, at several places, of thecommutation relation between an element Tx,y ∈ Pd and a rotation % ∈SO(d). We postpone the full commutation formula (we do not need) tothe appendix, see Section IX.2. According to the Iwasawa decomposition(Theorem I.7.2), there exists unique Tx′,y′ ∈ Pd and %′ ∈ SO(d) such that

% Tx,y = Tx′,y′ %′.

Theorem II.2.1 Denote by u(%) the Poincaré coordinate of

R+%(e0 + e1)(u(%) =∞ if and only if % ∈ SO(d− 1)

). We have :

(i) u(%′−1) =u(%−1)− x

yor equivalently (ii) u(%′) =

u(%)− x′y′

.

Moreover

(iii) Tx′,y′ e0 = % Tx,y e0 and (iv) |e1 − % e1|2 = 4/(|u(%)|2 + 1

).

The last assertion yields the interpretation :

u(%)→∞⇐⇒ % e1 → e1 ⇐⇒ %→ SO(d− 1).Proof (iii) is obvious, and determines x′, y′, by Proposition I.6.1.

II.2. A COMMUTATION RELATION 47

(i) By definition, we must have λ θ+u(%)(e0 − e1) = %(e0 + e1), for somereal λ . Then

2λ = λ〈e0 + e1, e0 − e1〉 =〈θ+−u(%)(e0 + e1), λ(e0 − e1)

〉=〈e0 + e1, λθ

+u(%)(e0 − e1)

〉= 〈e0 + e1, %(e0 + e1)〉 = 〈%(e0 + e1), e0 + e1〉.

Then, by Definition I.4.1, we have for 2 ≤ j ≤ d :

〈u(%), ej〉 = 12 〈e0 − e1, θ+−u(%)ej〉 = 12 〈θ+u(%)(e0 − e1), ej〉

=〈%(e0 + e1), ej〉

〈%(e0 + e1), e0 + e1〉.

On the other hand, for any v ∈ R1,d we have :

〈%′−1(e0 + e1), v〉 = y′ 〈T−1x′,y′(e0 + e1), %′v〉 = y′ 〈e0 + e1, Tx′,y′ %′v〉

= y′ 〈e0 + e1, % Tx,y v〉 = y′ 〈%−1(e0 + e1), Tx,y v〉.Hence we obtain :

u(%′−1) = −d∑j=2

〈u(%′−1), ej〉 ej = −d∑j=2

〈%′−1(e0 + e1), ej〉〈%′−1(e0 + e1), e0 + e1〉

ej

= −d∑j=2

〈%−1(e0 + e1), Tx,y ej〉〈%−1(e0 + e1), Tx,y(e0 + e1)〉

ej

= −d∑j=2

〈%−1(e0 + e1), xj(e0 + e1) + ej〉〈%−1(e0 + e1), y (e0 + e1)〉

ej =u(%−1)− x

y.

(ii) is deduced at once from (i), applied to %−1 Tx′,y′ = Tx,y %′−1 .

(iv) Finally we get from the above expression for u(%) :


〈u(%), ej〉 =〈% e1, ej〉

1 + 〈% e1, e1〉, whence :

(|u(%)|2 + 1)× (1 + 〈% e1, e1〉)2 = 1 + 2〈% e1, e1〉+ 〈% e1, e1〉2 +d∑j=2

〈% e1, ej〉2

= |e1 − % e1|2 = 2 (1 + 〈% e1, e1〉) . �See the proof of Lemma V.3.5 and its figure V.1 for a more intuitiveproof, in other coordinates and in two dimensions.

II.3 Flows and leaves

Recall that PSO(1, d) has a right action on the set Fd of Lorentzframes (recall Definition I.2.2 and Remark I.3.2). In particular, theright action of the subgroups (θt) and (θ

+u ) introduced in Definition I.4.1

defines the two fundamental flows acting on Lorentz frames.

Definition II.3.1 The geodesic flow is the one-parameter group definedon Fd by :

β 7→ β θt , for any β ∈ Fd and t ∈ R . (II.4)The horocycle flow is the (d− 1)-parameters group defined on Fd by :

β 7→ β θ+u , for any β ∈ Fd and u ∈ {e0, e1}⊥ ≡ Rd−1. (II.5)

Proposition II.3.2 The projection π0(β θt), of the orbit of a Lorentzframe β under the action of the geodesic flow, is a geodesic of Hd.Precisely, this is the geodesic determined by the plane {β0, β1}, and wehave :

d

dt(β θt)0 = (β θt)1 , and dist

(β0, (β θt)0

)= |t| .

II.3. FLOWS AND LEAVES 49

Proof The expression of θt yields at once : (β θt)0 = (ch t)β0+(sh t)β1 ,

and (β θt)1 = (sh t)β0+(ch t)β1 =d

dt(β θt)0 . Finally, by Proposition I.7.3

we have :

dist(β0, (β θt)0

)= argch

[〈β0, (β θt)0〉

]= argch [ch t] = |t| . �

Corollary II.3.3 The geodesic segment [p, p′] joining p, p′ ∈ Hd has length dist (p, p′), andis the unique minimising curve joining p to p′.

Proof The first claim follows at once from Proposition II.3.2 : we have necessarily [p, p′] ={(β θt)0

∣∣ 0 ≤ t ≤ dist (p, p′)}, for some Lorentz frame β . Then, if q belongs to someminimizing curve joining p to p′ = (β θr)0 in Hd, setting s := dist (p, q), we must have onone hand : [p, q] =

{(β′′θv)0

∣∣ 0 ≤ v ≤ s}, for some Lorentz frame β′′, and then q = (β % θs)0for some % ∈ SO(d), and on the other hand : dist (q, p′) = r − s , and[q, p′] =

{(β′ θu)0

∣∣ 0 ≤ u ≤ r − s}, for some Lorentz frame β′.This implies β′ = β % θs %

′ and β′ θr−s = β θr %′′ for some %′, %′′ ∈ SO(d), and then

θs %′ θr−s = %

−1 θr %′′. By Theorem I.7.1, we have er = ‖θr‖ = ‖θs %′ θr−s‖, which implies

%′(e0 + e1) = λ(e0 + e1), for some real λ , since R(e0 + e1) is the eigenspace of θr associatedwith its eigenvalue er. Since %′ ∈ SO(d), this forces λ = 1, and then %′ ∈ SO(d−1). Hence,we have θr %

′ = %−1 θr %′′, so that Theorem (I.7.1,(ii)) forces %, %′′ ∈ SO(d− 1) too, whence

q = (β θs)0 ∈ [p, p′]. �

Note that the geodesic flow makes sense also at the level of line ele-ments of Hd : recalling the identification (Hd× ∂Hd ≡ T 1Hd) of SectionII.1.2, we can set π1(β) θt := π1(β θt).

Indeed, if π1(β) = π1(β′), then % := β̃−1β̃′ ∈ SO(d − 1), so that π1(β′ θt) = π1(β % θt) =

π1(β θt %) = π1(β θt). Moreover, we have (βθt)0 + (βθt)1 = et (β0 + β1), for any real t .

Thus the action of the geodesic flow (θt) on the boundary componentof π1(β) ∈ Hd × ∂Hd is trivial, and θt moves the generic line elementalong the geodesic it generates, by an algebraic hyperbolic distance t .

On the contrary, the horocycle flow does not make sense at the level ofline elements (for d ≥ 3).


The geodesic determined by a given (p, pη) = J(p, η) = J ◦ π1(β) ∈ T 1Hd (using thenotations of Section II.1.2), with β ∈ Fd (determined up to the right action of SO(d − 1)),is parametrised by its real arc-length s as follows : s 7→ (ch s)β0 + (sh s)β1 .In other words, any geodesic of Hd is the isometric image (under β̃) of the geodesic run bythe point having Poincaré coordinates (0, es) (s being the arc-length). We can thus say thatit has Poincaré coordinates (0, es) in the frame β (instead of the canonical frame).

Remark II.3.4 Example : For an appropriate β ∈ Fd, the image under β̃−1 of a givenharmonic quadrangle Q (recall Definition II.1.3.3) has Poincaré coordinates {−1, 0, 1,∞}.Precisely, this means (recall Proposition II.1.2.4) that

β̃−1(Q) ={R+θ+−e2(e0 − e1),R+(e0 − e1),R+θ+e2(e0 − e1),R+(e0 + e1)

}.

Indeed, denoting the harmonic quadrangle by {η, η′′, η′, η′′′}, we can restrict to the 3-subspacecontaining it, and take : β0 to be the intersection of geodesics {η, η′} and {η′′, η′′′}, β1 :=(β0)η′′′ , β2 := (β0)η .

(Note that writing −1, 0, 1 means actually (−1, 0), (0, 0), (1, 0) ; this

is a usual writing, since no ambiguity can occur, as long as it is clear that ideal points (i.e.light rays) are considered.

)This extends Remark II.1.3.5 to d ≥ 2 .

Definition II.3.5 (i) Given η ∈ ∂Hd, a horosphere based at η is theintersection of Hd with an affine hyperplane (of R1,d) orthogonal to η .

(ii) Given a Lorentz frame β , let H(β) := Hd∩(β0+(β0+β1)

⊥)

denote

the horosphere based at R+(β0 + β1) and translated of the hyperplane(β0 + β1)

⊥ by β0 .

(iii) A horosphere H based at η determines the horoball H+, which isthe intersection of Hd with the closed affine halfspace of R1,d delimitedby H and containing η .

In analogy with the geodesic flow, concerning the horocycle flow we havethe following.

Proposition II.3.6 The horosphere through a Lorentz frame β is theprojection of its orbit under the action of the horocycle flow : H(β) =π0(β θ+Rd−1

). For any u ∈ Sd−2,

(β0 ,

dodε

(β θ+ε u)0

)belongs to T 1Hd.

II.3. FLOWS AND LEAVES 51

Moreover, we have

H(β) = H(β′)⇐⇒(∃ x ∈ Rd−1, % ∈ SO(d− 1)

)β = β′ θ+x % .

Proof By the expression of θ+u displayed in Definition I.4.1, we havefor any u ∈ Rd−1 :〈θ+u e0 , e0 + e1〉 = 1 , which is equivalent to :

〈(β θ+u )0−β0 , β0 +β1

〉= 0 ,

and then to π0(β θ+u ) ⊂ β0 + (β0 + β1)⊥. As π0(β θ+u ) is a connected

(d − 1)-dimensional submanifold of Hd ∩[β0 + (β0 + β1)

⊥], we mustindeed have π0(β θ

+u ) = Hd ∩

[β0 + (β0 + β1)

⊥]. Then, for u ∈ Sd−2 wehave : 〈

β0 ,dodε

(β θ+ε u)0

〉=dodε〈e0, θ+ε u e0〉 =

dodε

(1 + ε2|u|2/2) = 0 ,

and〈dodε

(β θ+ε u)0 ,dodε

(β θ+ε u)0

〉=

〈dodε

(θ+ε u e0),dodε

(θ+ε u e0)

〉= 〈u, u〉 = −1 .

Suppose H(β) = H(β′). According to Definition II.3.5, we have β0 =(β′ θ+x )0 for some x ∈ Rd−1. Hence, up to changing β′ into β′ θ+−x ,

we can suppose also β0 = β′0 . Set β

′1 =:

d∑j=1

λj βj , withd∑j=1

λ2j = 1 .

We must then have, for any u ∈ Rd−1 : (β θ+u )0 ∈ β0 + (β0 + β′1)⊥,

i.e. 0 =〈

12 |u|2(β0 + β1) + β(u) , β0 + β′1

〉= 12 |u|2(1− λ1)−

d∑j=2

λj uj ,

whence λ1 = 1 and λ2 = . . . = λd = 0 . Hence, β′1 = β1 , and then

β̃−1β̃′ ∈ SO(d− 1). The reciprocal is obvious, by Definition II.3.5. �

Mixing the actions of the two flows, we get the notion of stable leaf,as follows. See Figure II.2.


Definition II.3.7 For any light ray η , denote by Fd(η) the set of allframes β ∈ Fd pointing at η , i.e. such that β0 + β1 ∈ η , and callit the stable leaf associated to the light ray η . We shall also writeFd(R+(β0 + β1)

)=: Fd(β) the stable leaf containing β .

Proposition II.3.8 The flows act on each stable leaf Fd(η). Pre-cisely, Fd(β) is the orbit of β under the right action of the subgroupof PSO(1, d) generated by Pd ∪ SO(d− 1) :

Fd(β) ={β % θt θ

+u

∣∣∣ % ∈ SO(d− 1), t ∈ R, u ∈ Rd−1}={β Tz %

∣∣∣Tz ∈ Pd, % ∈ SO(d− 1)}.Proof We already noticed in Section I.4 that the lightlike vector (e0 + e1) ∈ R1,d is aneigenvector for each matrix θt θ

+u ∈ Pd. Owing to Definition II.3.1, this means exactly that

the flows act on each stable leaf Fd(η). The Iwasawa decomposition, applying Theorem I.7.2to γ−1, yields : PSO(1, d) =

{% θt θ

+u

∣∣ % ∈ SO(d), t ∈ R, u ∈ Rd−1}.Since % ∈ SO(d) fixes (e0 + e1) if and only if it belongs to SO(d − 1), this implies atonce that the subgroup of PSO(1, d) fixing the light ray R+(e0 + e1) ∈ ∂Hd is precisely{% θt θ

+u

∣∣ % ∈ SO(d − 1), t ∈ R, u ∈ Rd−1}. Whence the first characterization. The secondone follows at once, by Formula (I.15) and Proposition I.4.2. �

II.4 Structure of horospheres, Busemann function

Denote by Hη the family of all horospheres based at η ∈ ∂Hd (recallDefinition II.3.5).

For any p ∈ Hd, there exists a unique Hη(p) ∈ Hη such that p ∈ Hη(p).Note that Hη(p) = (p+η⊥)∩Hd, so that Hη(p) = Hη(p′)⇔ p−p′ ∈ η⊥ .

For any β ∈ Fd, we have H(β) = HR+(β0+β1)(β0).

II.4. STRUCTURE OF HOROSPHERES, BUSEMANN FUNCTION 53

d

d

Figure II.2: four frames of the stable leaf Fd(η) ≡ Fd(β) and two horospheres of Hη

Note that for any p ∈ Hd and any u = ξ + η ∈ η⊥/η , there existsa unique up ∈ {p, η}⊥ such that up ∈ u . Namely, up = (ξ + η)p :=ξ − 〈ξ, p〉 ηp .

Proposition II.4.1 Any H ∈ Hη is an affine space directed by η⊥/η ,the vector associated to (p, p′) ∈ H2 being p′ − p+ η .For any η ∈ ∂Hd, H ∈ Hη , p ∈ H, we have H + η⊥ = p+ η⊥ = H + η .

Proof For any H ∈ Hη , (p, p′) ∈ H2, and β ∈ Fd such that (β0, β1) = (p, pη), setting

x =d∑j=2

xj βj := (p′ − p+ η)p , i.e. xj := −

〈(p′ − p+ η)p, βj

〉,

we have β̃−1(x) ∈ Rd−1, x = p′ − p− 〈p′ − p, p〉 ηp = p′ − 〈p′, p〉 p+ (1− 〈p′, p〉)pη , and then|x|2 = −〈x, x〉 = −〈p′ − p, p′ − p〉 = 2(〈p′, p〉 − 1), whence

π0

(β ◦ θ+

β̃−1(x)

)= β̃

((1 + |x|

2

2

)e0 +

|x|22 e1 +

d∑j=2

xj ej

)= 〈p′, p〉 p+ (〈p′, p〉 − 1)pη + x = p′.


This proves the first assertion. The first equality of the second sentence of the statementfollows at once from H = Hd∩(p+η⊥), which holds by Definition II.3.5. Then fix β ∈ Fd(η)such that H = H(β). Any ξ ∈ η⊥ can be written

ξ = α(β0 + β1) +d∑j=2

ujβj , for some α ∈ R

hyperbolic dynamics - université paris-saclaylejan/pdfs/... · 2011. 2. 28. · iii.5) the...

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