remarks on the hyperbolic geometry of...

NEW ZEALAND JOURNAL OF MATHEMATICSVolume 35 (2006), 85–107

REMARKS ON THE HYPERBOLIC GEOMETRY OF PRODUCTTEICHMULLER SPACES

Ioannis D. Platis

(Received February 2003)

Abstract. Let eT be a cross product of n Teichmuller spaces of Fuchsian groups,

n > 1. From the properties of Kobayashi metric and from the Royden-Gardiner

theorem, eT is a complete hyperbolic manifold. Each two distinct points of eTcan be joined by a hyperbolic geodesic segment, which is not in general unique.

But when eT is finite dimensional or infinite dimensional of a certain kind,

then among all such segments there is only one which enjoys a distinguishedproperty: it is obtained from a uniquely determined holomorphic isometry of

the unit disc into eT .

If QC(G) is the Quasiconformal Deformation space of a finitely generated

Kleinian group G, then since its holomorphic covering is a product of finitedimensional Teichmuller spaces, all the above results hold for QC(G).

1. Introduction and Statement of Results

Our aim is to study basic aspects of the hyperbolic geometry of products ofTeichmuller spaces of Fuchsian groups. This is the geometry obtained by endowingthese spaces with the Kobayashi-hyperbolic metric.

Let Γ be a Fuchsian group acting on the upper half plane and denote by Teich(Γ)its Teichmuller space. Teich(Γ) is a complex manifold and admits various natu-ral metrics. The metric we are interested in is the Teichmuller metric τ

Twhich

measures the distance between two points in terms of maximal dilatations of quasi-conformal deformations of Γ. This notion was first introduced by O. Teichmuller inthe special case where Γ is of signature (g, 0), that is when Γ uniformises a closedRiemann surface of genus g > 1. This metric, later defined for Teichmuller spacesof all kinds of Fuchsian groups, is a complete non-Riemannian metric.

To each complex manifold M is assigned a generally non-Riemannian pseudo-metric, the Kobayashi pseudometric κ

M, which is such that its group of isometries

is exactly the group of biholomorphic automorphisms of M. When κM

is a metric,M is called a hyperbolic manifold.

The Teichmuller metric is intimately related to the Kobayashi metric and thusto the hyperbolic geometry of Teich(Γ). H. Royden proved in 1971 that τ

T= κ

T

in the case where Γ is of the first kind and of signature (g, 0), g > 1 [15]. Later,this result was extended by F. Gardiner to the case where Γ is arbitrary (Chapter7 of [5]) and has been known ever since as Royden-Gardiner Theorem (RGT).According to RGT, Teich(Γ) is a complete hyperbolic manifold and moreover, when

1991 Mathematics Subject Classification Primary 30F40, Secondary 30F60.Key words and phrases: Teichmuller spaces, Kleinian groups, Teichmuller metric, Kobayashi met-ric, hyperbolic manifolds, geodesics.

86 IOANNIS D. PLATIS

Γ is of the first kind (with the exception of a few cases), the set of biholomorphicautomorphisms of Teich(Γ) is just Mod(Γ), the Modular group of Γ. (The casewhere Γ is of signature (g, n), 2g + n > 4, was proved by Earle and Kra. [3] Thecase where Teich(Γ) is infinite dimensional is more complicated, (see §2 below).

Let now T be the cross product of the Teichmuller spaces Teich(Γi) of Fuchsiangroups Γi, i = 1, ..., n for n > 1. From the product property of Kobayashi metric[7] we deduce that T is a hyperbolic manifold and

κ eT = maxi=1,..,n

κT (Γ

i) = max

i=1,..,nτ

T (Γi)

where τT (Γ

i) is the Teichmuller metric in Teich(Γi), i = 1, ..., n.

The Teichmuller metric τ eT can be therefore defined in a natural way as

τ eT = maxi=1,..,n

τT (Γ

i).

In this manner the most general version of RGT can be stated as below.

Theorem 1.1. Let Γ1 , ...,Γn be Fuchsian groups and T be the cross product of theirTeichmuller spaces. Then T is a complete hyperbolic manifold: the Teichmuller andKobayashi metrics coincide in T .

It is worth remarking that there are no restrictions for the kind of Fuchsiangroups that we consider here.

According to the RGT, holomorphic isometries of (T , τ eT ) are just the holomor-phic automorphisms of T . When all Γ

iare of the first kind these automorphisms

have been studied extensively by J. Gentilesco in [6]. Since the universal coveringspace of the Quasiconformal Deformation space QC(G) of a Kleinian group G is across product of Teichmuller spaces, he is able to state some results about the groupof automorphisms of this deformation space. In particular, it is not a homogeneousspace. Our Theorem 4.2 in §2.3 follows from the generalised version of RGT andTheorem III in [6].

The main part of our work is devoted to the study of holomorphic isometriesand geodesics of the hyperbolic manifolds we are concerned with. The model forthese manifolds is the hyperbolic unit polydisc ∆n of Cn, n > 1. Given two distinctpoints in ∆n there is always a geodesic (with respect to the metric) segment joiningthem. In fact, one can prove that any two such points can be joined by infinitelymany geodesic segments, which all lie in a closed region of ∆n. Therefore, thereis no hope of obtaining a unique (in the classical sense) segment joining any twopoints of T even in the finite dimensional case.

We find our way out of this difficulty by observing that among all geodesicsegments which join two points of the polydisc there is only one characterised by aspecial property: it can be obtained by a uniquely determined holomorphic isometryof the disc into the polydisc. (See Proposition 3.4). Not surprisingly, in this caseit coincides with the segment we take when ∆n is equipped with the standardBergman metric.

We call this segment maximal, and the geodesic curve upon which it lies amaximal geodesic. Our further study shows that the above picture in the polydiscappears in the class of manifolds we study. Combining our general results withclassical statements concerning Teichmuller space we obtain our main result

HYPERBOLIC GEOMETRY OF PRODUCT TEICHMULLER SPACES 87

Theorem 1.2. Let T be a cross product of Teichmuller spaces Teich(Γi) of Fuch-

sian groups Γi. For any two distinct points φ and ψ of T there is a geodesic (with

respect to Teichmuller-Kobayashi metric) segment joining them. If T is finite di-mensional then there is a maximal geodesic segment joining φ and ψ. The sameholds also in the case where T is infinite dimensional and of a special form.

Furthermore, T has a straight space property: maximal geodesics are isometricwith the Euclidean real line. We finally wish to remark that as applications toour results here are in the study of the hyperbolic geometry of Moduli spaces ofKleinian Groups. Here we just make a hint in §5.

This work has been carried out while the author was visiting the Department ofMathematics of the University of Crete, Greece.

2. Preliminaries

Definitions and results stated in this section are standard. For instance, werefer the reader to [10] and [13] for an extensive presentation of Teichmuller andDeformation space theory respectively.

2.1. Teichmuller space.

2.1.1. Definition of Teichmuller space.Let Γ be a Fuchsian group acting on the upper half plane U. We denote by

L∞(Γ) the space of invariant differentials for Γ, that is the complex Banach space ofmeasurable complex essentially bounded functions µ(z) defined on U and satisfyingthe transformation law

µ(γ(z))γ′(z)/γ′(z) = µ(z)

for all γ ∈ Γ and all z ∈ U. Belt(Γ) is the space of Beltrami differentials for Γ, thatis the open unit ball of L∞(Γ).

Let µ ∈ Belt(Γ) and denote by fµ the unique homeomorphism solution (nor-malised so that it fixes 0, 1 and ∞) to the Beltrami equation

fz

=µ(z)f

z

µ(z)fz

z ∈ Uz ∈ C \ U.

The map fµ is quasiconformal; an elementary argument shows that fµΓ(fµ)−1 isagain a Fuchsian group.

Definition 2.1. Teichmuller space Teich(Γ) is the space of equivalence classesof elements of Belt(Γ), where two such differentials µ, ν are equivalent if fµ, fν

coincide on the real axis.

2.1.2. Complex structure and Bers’ embedding.Teich(Γ) is a complex manifold modelled on a complex Banach space with

dimC(Teich(Γ)) = +∞ unless Γ is finitely generated and of the first kind. ThenTeich(Γ) is isomorphic to the Teichmuller space Teich(Γ′) where Γ′ is a Fuchsiangroup of signature (g, n) : the resulting Riemann surface U/Γ′ is analytically finite,i.e of genus g > 1 with n punctures. In this case, dimC(Teich(Γ)) = 3g − 3 + n.

The complex manifold structure of Teichmuller space arises from the Bers’ em-bedding which identifies it with an open subset of the complex Banach space B(Γ),


the space of bounded quadratic differentials for Γ in the lower half plane L. Anelement φ ∈ B(Γ) is a bounded holomorphic function φ(z) on L, that is

‖ φ ‖∗ = supφ(z)y2

, z ∈ L<∞

satisfying the transformation law

φ(γ(z))(γ′(z))2 = φ(z)

for all γ ∈ Γ and for all z ∈ L.Teichmuller space may be identified with a subset of B(Γ) which contains the

ball B(0, 2) and is contained in the ball B(0, 6). From now on, Teich(Γ) will be thissubset .

The mappingΦ : Belt(Γ) → Teich(Γ)

sending each µ to the corresponding φ is holomorphic and is called the Bers’ pro-jection.

2.1.3. Modular group.We consider the group M(Γ) of all quasiconformal mappings h of U such that

h γ h−1 ∈ Γ for all γ ∈ Γ and the group M0(Γ) of all quasiconformal mappingsg of U such that g γ g−1 = γ for all γ ∈ Γ. The Modular group Mod(Γ) isthe quotient M(Γ)/M0(Γ). There is a homomorphism of Mod(Γ) into the group ofbiholomorphic automorphisms Aut(Teich(Γ)) of Teich(Γ)); if h is a representativeof a coset, then this coset is mapped into the automorphism γ

hwhich is such that

for every µγ

h(Φ(µ)) = Φ(ν)

where ν is the Beltrami differential of fµ h−1. Following [2] we shall call suchautomorphisms geometric. The group of geometric automorphisms shall be denotedby Geom(Γ).

2.1.4. Extremal differentials and Teichmuller metric.Teichmuller’s Existence Theorem for extremal quasiconformal mappings of the

surface U/Γ states that among all quasiconformal mappings of U/Γ which belong tothe same class in Teich(Γ) there exists one that is extremal, that is of the smallestmaximal dilatation. If fµ is this mapping then µ satisfies

‖ µ ‖∞= inf‖ ν ‖∞ ; ν ∈ Belt(Γ), Φ(µ) = Φ(ν)

Such a µ is also called extremal.The Teichmuller metric is defined as follows. In the space of Beltrami differentials

Belt(Γ) define the Teichmuller metric τB(Γ) by

τB(Γ)(µ, ν) = tanh−1

∥∥∥∥ µ− ν

1− µν

∥∥∥∥∞

for every µ, ν in Belt(Γ). Then the Teichmuller metric τT (Γ) of Teich(Γ) is given

byτ

T (Γ)(ϕ,ψ) = infτB(Γ)(µ, ν); Φ(µ) = ϕ, Φ(ν) = ψ

for every ϕ,ψ in Teich(Γ).


It follows from Teichmuller’s Existence Theorem that τT (Γ) is a well defined

metric. We also note that τT (Γ) is invariant by the group Geom(Γ) of geometric

automorphisms of Teich(Γ).A µ ∈ Belt(Γ) is called uniquely extremal if

Φ(µ) 6= Φ(ν)

for every ν ∈ Belt(Γ) such that ν 6= µ and ‖ ν ‖∞≤‖ µ ‖∞ . A uniquely extremalBeltrami differential is also extremal. If dimC(Teich(Γ)) < +∞, every extremalBeltrami differential is unique in its class due to Teichmuller’s Uniqueness Theoremand thus uniquely extremal. It belongs to Belt

T(Γ), the space of Teichmuller

differentials for Γ. This space contains differentials of the form

µ = λ| φ |φ, λ ∈ [0, 1), φ ∈ Q(Γ),

where Q(Γ) is the set of holomorphic quadratic differentials for Γ, that is integrableholomorphic functions in U which satisfy the transformation law:

φ(γ(z))(γ′(z))2 = φ(z)

for all γ ∈ Γ and for all z ∈ U.In the infinite dimensional case, an extremal µ is not necessarily uniquely ex-

tremal [11]. But there, uniquely extremal Beltrami differentials µ with the addi-tional property

‖ µ ‖∞=| µ |almost everywhere, are of particular importance.

2.1.5. Royden-Gardiner theorem.The proof of Royden-Gardiner Theorem (RGT) given in [4] is based on the

following Lifting Theorem.

Theorem 2.2 (Lifting Theorem). If f : ∆ → Teich(Γ) is a holomorphic mappingof the unit disc ∆ into Teich(Γ), then there exists a holomorphic mapping g : ∆ →Belt(Γ) such that

Φ g = f.

If µ0 ∈ Belt and Φ(µ0) = f(0), we can choose g so that g(0) = µ0 .

Theorem 2.3. (RGT) The Teichmuller and Kobayashi metrics of Teich(Γ) coin-cide.

(For the definition and properties of Kobayashi metric see §3.) Combining RGTand results stated in [2] we have the following.

Theorem 2.4.(a) Let dimC(Teich(Γ) < +∞. Then Mod(Γ) is identified to the group Geom(Γ)

of geometric automorphisms of Teich(Γ) and the latter is the full group ofbiholomorphic isometries I(Teich(Γ)) of Teich(Γ) with respect to τ

T (Γ) . Thus,it is the full group Aut(Teich(Γ)) of biholomorphisms of Teich(Γ). (A fewcases are excluded, see the remark following Theorem 2.12 below).


(b) Let dimC(Teich(Γ) = +∞. Then Geom(Γ) is identified to Aut(Teich(Γ)) (andthus to I(Teich(Γ)) if Γ has the isometry property: If Q(S) is the space ofintegrable holomorphic quadratic differentials of U/Γ, and X,Y are Riemannsurfaces quasiconformally equivalent to S, then every complex linear isometryof Q(X) onto Q(Y ) is of the form

φ → af∗(φ)

where a ∈ C, f is a holomorphic isomorphism of Y onto X and f∗ is the pull-back mapping induced by f . Also, if Γ has the isometry property, an extremalµ is a Teichmuller differential.

2.2. Quasiconformal deformation space of a Kleinian group.Let G be a finitely generated, non elementary and torsion free Kleinian group

with region of discontinuity Ω(G) and limit set Λ(G). Then,

Ω(G)/G =n⋃

i=1

Si

where Si , for i = 1, ..., n are analytically finite Riemann surfaces: each Si is uni-formised by a Fuchsian group Γ

iof signature (g

i, n

i) acting on the upper half plane.

The collection Γ1 , ...,Γn is called the Fuchsian model of G in Ω(G).

Definition 2.5. The Teichmuller space of the Kleinian groupG is the cross product

T (G) =n∏

i=1

Teich(Γi)

where Γi, for i = 1, ..., n is the Fuchsian model for G.

Denote by L∞(G) the space of invariant differentials for G, that is the complexBanach space of measurable, complex, essentially bounded functions µ(z) definedon C with support on Ω(G) and satisfying the transformation law:

µ(γ(z))γ′(z)/γ′(z) = µ(z)

for all γ ∈ G and all z ∈ C. The open unit ball Belt(G) of L∞(G) is the space ofBeltrami differentials for G.

Let µ ∈ Belt(G) and denote by wµ the unique homeomorphism solution (nor-malised so that it fixes 0, 1 and ∞) to the Beltrami equation

wz

= µ(z)wz

in C. wµ is quasiconformal and wGw−1 is again a Kleinian group.

Definition 2.6. Quasiconformal Deformation space QC(G) of G is the set ofequivalence classes of Beltrami differentials, where two such differentials µ, ν areequivalent if wµ, wν coincide on the limit set of G.

QC(G) admits a natural complex structure. The natural surjection

Φ : Belt(G) → QC(G)

given byΦ(µ) = [µ]

for each µ ∈ Belt(G), is holomorphic.


The complex manifold structure of QC(G) is given by Bers’ Theorem [1], [8],[13]:

Theorem 2.7 (Bers’ Theorem). If G is a finitely generated, torsion free Kleiniangroup with Fuchsian model Γ1 , ...,Γn

then there exists a holomorphic covering

Ψ : T (G) → QC(G)

with holomorphic covering group a fixed point free subgroup L1×...×Lnof Mod(Γ1)×

...×Mod(Γn). The mapping

ΨG

: Teich(Γ1)/L1 × ...× Teich(Γn)/Ln → QC(G)

is biholomorphic and QC(G) is a simply connected complex manifold.

Note.(a) Denote by pr

ithe natural projections Teich(Γ

i) → Teich(Γ

i)/L

i, i = 1, ..., n.

ThenΨ = Ψ

G pr

where pr = (pr1 , ..., prn).

(b) If Ω(G) consists of simply connected components then Ψ is a biholomorphism.

2.3. Kobayashi metric and hyperbolic manifolds.For the following definition and results concerning the Kobayashi metric we

follow [4]. We also refer to [9] and [7] for further details.Let M be a complex open Banach manifold and denote by H(∆,M) the set of

holomorphic mappings from the unit disc into M. The Kobayashi function

δM

: M ×M → [0,+∞]

is defined for every p, q ∈M by

δM

(p, q) = infd∆(0, t)where the infimum is taken over all f ∈ H(∆,M) such that f(0) = p and f(t) = q.

If M,N are complex Banach manifolds and f : M → N is a holomorphic map-ping, then

δN

(f(p), f(q)) ≤ δM

(p, q)for every p, q ∈M. The equality holds if f is biholomorphic.

Definition 2.8. The Kobayashi (pseudo)metric κM

is the largest (pseudo)metricon M such that

κM

(p, q) ≤ δM

(p, q)for all p, q ∈M .

If δM

is a metric then δM

= κM

and M is called a hyperbolic manifold. If κM

iscomplete, then M is a complete hyperbolic manifold.

It is clear that(a) the Kobayashi metric is distance decreasing: if M,N are hyperbolic manifolds

and f : M → N is a holomorphic mapping, then

κN

(f(p), f(q)) ≤ κM

(p, q)

for every p, q ∈M.


(b) The group of holomorphic isometries I(M) of a hyperbolic manifold M , is justthe group Aut(M) of its biholomorphic automorphisms.

The following also holds [7].

Theorem 2.9. Let Mi, i = 1, ..., n be hyperbolic manifolds and M be their cross

product. Then M satisfies the product property: for each p = (p1 , ...pn) and q =

(q1 , ..., qn) in M we have

κfM (p, q) = maxi=1,...,n

κM

i(p

i, q

i).

Denote by ∆ the unit polydisc of Cn

z = (z1 , ..., zn) ∈ Cn :| z

i|< 1, i = 1, ...n.

Thenκ e∆(z, w) = d e∆(z, w) = max

i=1,...,nd∆(z

i, w

i)

(cf. [9] p. 47). The metric d e∆ coincides with the Bergman metric defined for ∆only in the case when ∆ = ∆.

We shall also need ([9] Proposition 1.6 and Theorem 4.7):

Theorem 2.10. Let M ′ and M be complex manifolds and π : M ′ → M be aholomorphic covering. Then(a)

κM

(p, q) = infκM′ (p′, q′)

where the infimum is taken over all p′, q′ ∈M ′ such that π(p′) = p and π(q′) =q and

(b) M ′ is (complete) hyperbolic if and only if M is (complete) hyperbolic.

Theorems 1.1 and 3.2 have immediate consequences to the manifolds we study.Let Γ1 , ...,Γn

be Fuchsian groups acting on the upper half plane,

B =n∏

i=1

Belt(Γi)

be the cross product of the spaces of their Beltrami differentials and

T =n∏

i=1

Teich(Γi)

be the cross product of their Teichmuller spaces. RGT and Theorem 1.1 imply

Theorem 2.11. B and T are complete hyperbolic manifolds:

κ eB (µ, ν) = maxi=1,..n

κB(Γ

i)(µi

, νi) = max

i=1,..nτ

B(Γi)(µi

, νi)

for every µ = (µ1 , ..., µn), ν = (ν1 , ..., νn) ∈ B and

κ eT (φ, ψ) = maxi=1,...,n

κT (Γ

i)(φi

, ψi) = max

i=1,...,nτ

T (Γi)(φi

, ψi)

for every φ = (φ1 , ..., φn), ψ = (ψ1 , ..., ψn) elements of T . Furthermore,

κ eT (φ, ψ) = infκ eB (µ, ν) : Φ(µ) = φ, Φ(ν) = ψ.


Suppose that T is finite dimensional and denote by M the cross productn∏

i=1

Mod(Γi) =

n∏i=1

Geom(Γi)

of the Modular groups of the Fuchsian groups Γi.

The following may be immediately deduced from Theorem 3 and a result dueto Gentilesco ([6], Theorem III). It describes the full group of holomorphic selfisometries I(T ) of (T , τ eT ) in the finite dimensional case.

Theorem 2.12. M acts properly discontinuously as a subgroup of holomorphicisometries of (T , κ eT ) when T is finite dimensional. If the signature (g

i, n

i) of each

Γi satisfies 2gi +ni > 4 then I(T ) and thus Aut(T ), is the semi direct product of Mby the finite subgroup H, where H is generated by elements h

ijdefined as follows:

let fij

: Teich(Γi) → Teich(Γ

j) be a chosen biholomorphic map. (Such a map

exists if Γi,Γ

jare of the same signature). Then h

ij= (k1 , ..., kn

) where ki

= fij,

kj

= f−1ij

and kr

= id for r 6= i, j.

Remark 2.13. It is worth discussing the exceptional cases to the above theorem.When Γ

iis of signature (2, 0), I(Teich(Γ

i)) ' Mod(Γ

i)/Z2 . If Γ

iis of signature

(0, 3) then Teich(Γi) is a single point. If it is of signature (1, 2) then Teich(Γ

i) is

biholomorphic to a Teichmuller space of a group of signature (0, 5). Finally, if Γi isof signature (1, 1) or (0, 4), then its Teichmuller space is conformally equivalent tothe upper half plane. For details, see [3].

We conjecture that an analogous result holds in the infinite dimensional case atleast when all Γi have the isometry property (see §2).

Conjecture 1. Let Γi, for i = 1, ..., n be Fuchsian groups which have the isometry

property and T be the cross product of their Teichmuller spaces. Also let

G =n∏

i=1

Geom(Γi)

be the cross product of the groups of their geometric automorphisms. Then thegroup of biholomorphic isometries I(T ) of T with respect to the Kobayashi metricis the semi-direct product of G by the finite subgroup H, where H is generated byelements hij defined as in Theorem 4.2.

We believe that this conjecture can be proved following the line of the proof ofTheorem III in [6].

Now let G be a finitely generated, torsion free and non elementary Kleinian groupwith non empty region of discontinuity. Let Γ1 , ...,Γn be its Fuchsian model and

T (G) =n∏

i=1

Teich(Γi)

be its Teichmuller space. By Theorem 3, T (G) is a complete hyperbolic manifold.Additionally, the group I(T (G)) is described in the following.


Theorem 2.14. Let (T (G), κ eT ) as before and M(G) be the cross productn∏

i=1

Mod(Γi) =

n∏i=1

Geom(Γi)

of the Modular groups of the Fuchsian groups Γi . If the type (gi , ni) of each Γi

satisfies 2gi+n

i> 4 then I(T (G)) is the semi-direct product of M(G) by the finite

subgroup H, where H is generated by elements hij

as in Theorem 4.2.

By Bers’ Theorem, QC(G) is a complex manifold, thus admits a Kobayashipseudometric. Moreover, Theorems 1.1, 3.2 and 4.1 immediately deduce

Theorem 2.15. Let QC(G) be the Quasiconformal Deformation space of a finitelygenerated, torsion free and non elementary Kleinian group G with Fuchsian modelΓ1 , ...,Γn. If T (G) is the Teichmuller space of G with hyperbolic metric κ eT andκ

Qis the Kobayashi pseudometric on QC(G), then for each [µ], [ν] ∈ QC(G),

κQ([µ], [ν]) = infκ eT (φ, ψ)

where the infimum is taken over all φ, ψ ∈ T (G) such that Ψ(φ) = [µ], Ψ(ψ) = [ν]and Ψ is the holomorphic covering T (G) → QC(G) given by Bers’ Theorem. Thus,QC(G) is a complete hyperbolic manifold. Moreover, if the region of discontinuityΩ(G) consists of simply connected components then its complete hyperbolic metricκ

Qis given for each [µ], [ν] ∈ QC(G) by

κQ([µ], [ν]) = κ eT (Ψ−1([µ]),Ψ−1([ν])).

3. Holomorphic Isometries and Geodesics of Hyperbolic Manifolds

3.1. Holomorphic isometries.Let M,N be hyperbolic manifolds and H(N,M) be the set of holomorphic map-

pings of N into M.

Definition 3.1. A mapping f ∈ H(N,M) is called a holomorphic isometry if forevery t, s ∈ N

κM

(f(t), f(s)) = κN

(t, s).We denote the set of holomorphic isometries by I(N,M).

If N = M, then the holomorphic isometries are exactly the biholomorphisms ofM. To describe completely the group I(M) of a hyperbolic manifold M is not atrivial task at all. The following classical result concerning the special case of thepolydisc will be needed later ([14], Proposition 3, p.68).

Theorem 3.2. Let (∆, d e∆) be the hyperbolic manifold

z = (z1 , ..., zn) ∈ Cn :| z

i|< 1, i = 1, ...n

with hyperbolic metric

d e∆(z, w) = maxi=1,...,n

d∆i(z

i, w

i).

Then for each F ∈ I(∆) there exists a permutation

σ : (1, ..., n) → (1, ..., n)


of the integers from 1 to n, real numbers θ1 , ..., θn, and complex numbers a1 , ..., an

,| a

i|< 1 for each i = 1, ..., n, such that

F (z1 , ..., zn) =

(eiθ1

zσ(1) − a1

1− a1zσ(1)

, ..., eiθnz

σ(n) − an

1− anz

σ(n)

).

Proposition 3.3. Let Mii = 1, ..., n be hyperbolic manifolds and M be their cross

product with hyperbolic metric

κfM = maxi=1,...,n

κM

i.

Then:(a) A mapping f : ∆ → M, where

f(z) = (f1(z), ..., fn(z))

belongs to I(∆, M) if each fi ∈ I(∆,Mi).

(b) If fi∈ I(∆,M

i) then F : ∆ → M defined by

F (z) = (f1(z1), ..., fn(z

n))

belongs to I(∆, M).

Proof. (a) Since each fi is a holomorphic isometry, we have that for everyt, s ∈ ∆

κM

i(f

i(t), f

i(s)) = d∆(t, s).

Now

κM

(f(t), f(s)) = maxi=1,...,n

κM

i(f

i(t), f

i(s))

= d∆(t, s),

which proves a).To prove (b) consider arbitrary z, w ∈ ∆. Then

κfM (F (z), F (w)) = maxi=1,...,n

κM

i(f

i(z

i), f

i(w

i))

= maxi=1,...,n

d∆i(z

i, w

i)

= d e∆(z, w).

We shall use the above results to describe the set I(∆, ∆). In the first place,Proposition 3.6 (a) induces that this set contains elements of the form

f(z) =(f1(z), ..., fn

(z))

where fi

are holomorphic automorphisms of the disc, i = 1, ..., n. If we also requiref(0) = 0, then from Schwarz’s Lemma we obtain

f(z) = (eiθ1 z, ..., eiθn z).


We shall now track down all the other elements f = (f1 , ..., fn) of I(∆, ∆) which

satisfy the condition f(0) = 0. Since all fi

are holomorphic self-mappings of thedisc and f

i(0) = 0, Schwarz’s Lemma implies

| fi(z) | ≤ | z | .

But there is at least one z0 ∈ ∆, z0 6= 0, such that | fj (z0) |=| z0 | for some j,otherwise f would not be a holomorphic isometry. Thus, again from Schwarz’sLemma,

fj (z) = eiθz

for at least this j. Take any other fi

which satisfies

| fi(z) |<| z |

and let mi be the order of the root 0 of fi , mi ≥ 1. Then we can write

fi(z) = zm

ihi(z)

for some holomorphic function of the disc hi, with h

i(0) 6= 0. We consider an

arbitrary but fixed z ∈ ∆, and an r ∈ (0, 1) such that | z |< r. Then hi(z) is

holomorphic in the disc | z |< r and continuous in the closed disc | z |≤ r.Applying the Maximum Principle, we have∣∣∣∣fi

(z)zm

i

∣∣∣∣ = | hi(z) |≤ max|ζ|=r

∣∣∣∣fi(ζ)ζm

i

∣∣∣∣ < 1rm

i−1 .

If mi= 1, then | h

i(z) |≤ 1. If m

i> 1 we let r → 1 to obtain

| hi(z) | =

∣∣∣∣fi(z)zm

i

∣∣∣∣ ≤ 1

where equality holds only iff

i(z) = c

izm

i

where | ci |= 1 (mi > 1). Therefore fi(z) = zmihi(z) is either of the above form or

such that | hi(z) |< 1.Note here that from the Pick’s formulation of Schwarz’s Lemma, we also have

d∆(hi(0), hi(z)) ≤ d∆(0, z).

Finally, if f(0) = z 6= 0 then f = F−1 g, where F is an automorphism ofthe polydisc mapping z to 0 (cf. Theorem 7) and g is an isometry of the abovedescribed form.

We conclude from the above discussion that in general, given two points z and win the polydisc with a = tanh d e∆(z, w) there can be a whole family of holomorphicisometries of the disc into the polydisc which map 0 to z and a to w. The followingproposition states that there is a unique such isometry which satisfies two additionalconditions.

Proposition 3.4. Let z = (z1 , ..., zn), w = (w1 , ..., wn

) be any two distinct pointsin ∆ and a = tanh d e∆(z, w), ai = tanh d∆(zi , wi), i = 1, ..., n. There exists a uniquef ∈ I(∆, ∆), f = (f1 , ..., fn), which satisfies the following conditions:(A) f(0) = z and f(a) = w.

(B) The image fi([0, a]) is the geodesic segment of ∆ joining z

iand w

i, i = 1, ..., n.


(C) For each i such that ai< a the quantities d∆(z

i, f

i(a

i)) are maximal in the

sense that if there exists a g = (g1 , ..., gn) in I(∆, ∆) different from f and

satisfying conditions (A) and (B) then

d∆

(z

i, f

i(a

i))> d∆

(z

i, g

i(a

i))

unless fi(a

i) = g

i(a

i).

Proof. For simplicity, we shall omit the exceptional case where zi

= wi

for somei. Suppose first that z = 0 = (0, ..., 0) and w = a = (a1 , ..., an

). We distinguish twocases:

(a) a1 = ... = an

= a. According to our previous discussion, the only f =(f1 , ..., fn) in I(∆, ∆) which satisfies condition (A) is f(z) = (z, ..., z).

Indeed, if fj(z) = eiθz for some j, then f

j(a) = a implies that f

j(z) = z. If now

fi(z) = zm

ihi(z) for some i, consider first the case where | h

i(z) |< 1.

Then fi(a) = a implies

ami−1hi(a) = 1

which is impossible. If fi(z) = c

izm

i then fi(a) = a implies c

i= 1 and m

i= 1

which is also a possibility that cannot occur. Thus for all i, fi(z) = z and therefore

f clearly satisfies condition (B). Condition (C) is vacuous in this case.(b) a

i6= a

jfor at least two different i, j. We claim that f is given by

f(z) =(a1

az,a2

az, ...,

an−1

az,an

az).

Clearly f ∈ I(∆, ∆) and satisfies conditions (A) and (B). We check that it is theonly one which also satisfies condition (C).

Let g = (g1 , ..., gn) ∈ I(∆, ∆) be different from f and such that it satisfies

conditions (A) and (B). Then, at least one gi

is the identity and all others are ofthe form

gi(z) = c

izm

i

where ci

is of modulus 1 and mi> 1 or is of the form

gi(z) = zmihi(z)

where mi> 0, h

iis a holomorphic function of the disc, h

i(0) is not 0 and | h

i(z) |<

1.In the first case condition (B) implies that c

i= 1 and from g

i(a) = am

i = ai

weobtain m

i= (lg a

i)/(lg a). Now m

i> 2 implies a

i> a2 and thus

gi(a) = am

i < a2 < ai

which is impossible. Thus mi= 2, a

i= a2 and g

i(z) = z2.

We now haved∆(0, g

i(a

i)) = d∆(0, a2

i) = d∆(0, a4)

On the other hand,

d∆(0, fi(ai)) = d∆(0, a2i/a) = d∆(0, a3)

and we are done with this case.Suppose now that g

i(z) = zm

ihi(z) as above. Then from condition (A) and

since hi

is bounded above by 1, we have

hi(a) =ai

ami< 1.


Thus ai< am

i and moreover, hi(t) is strictly increasing in [0, a], therefore h

i(a

i) <

hi(a). Now,

d∆(0, gi(ai)) = d∆(0, ami

ihi(ai))

< d∆(0, ami

ihi(a))

< d∆(0, (ai/a)m

iai)< d∆(0, (ai/a)ai)= d∆(0, fi(ai))

and we are done in all cases.Let now z, w be arbitrary, and a

i= tanh d∆(z

i, w

i). There exists an automor-

phism F = (F1 , ..., Fn) of the polydisc, mapping z to 0 and w to a = (a1 , ..., an).Namely,

F (ζ1 , ..., ζn) =

(eiθ1

ζ1 − z1

1− z1ζ1, ..., eiθn

ζn− z

n

1− znζ

n

)where

eiθi = a

i

1− ziwi

wi− z

i

.

Set f = F−1 f0 where f0 is the unique element of I(∆, ∆) obtained in a). Thenf clearly satisfies conditions (A) and (B).

Let g ∈ I(∆, ∆), g = (g1 , ..., gn) different from f which also satisfies conditions(A) and (B). Suppose further that there exists an i such that

d∆

(zi , fi(ai)

)≤ d∆

(zi , gi(ai)

)and f

i(a

i) 6= g

i(a

i). Since f

i= F−1

i (f0)i

and Fi

is a holomorphic isometry of thedisc, by applying F

ito the above inequality we obtain

d∆

(0, (f0)i(ai)

)≤ d∆

(0, (Fi gi)(ai)

).

for some i. This can not happen unless (f0)i = Fi gi for all i. But then gi =F−1

i (f0)i = fi and therefore fi(ai) = gi(ai) which is a contradiction. Thus such

a g can not exist and therefore f satisfies condition (C).

3.2. Geodesics.

Definition 3.5. Let M be a hyperbolic manifold with Kobayashi metric κM

andc : [a, b] →M be a continuous mapping. We call the image J = c([a, b]) a geodesicsegment [p, q] joining the endpoints c(a) = p, c(b) = q of J if for every t1 ≤ t2 ≤t3 ∈ [a, b] the following holds:

κM

(c(t1), c(t3)

)= κ

M

(c(t1), c(t2)

)+ κ

M

(c(t2), c(t3)

).

It is unique if for every r ∈M not in [p, q] we have

κM

(p, r) + κM

(r, q) > κM

(p, q).

A geodesic curve γ is the image of a continuous mapping c : (−∞,+∞) →M suchthat the image of every closed subinterval is a geodesic segment.

The next proposition states that holomorphic isometries map geodesic segments(resp. curves) to geodesic segments (resp. curves).


Proposition 3.6. Let M,N be hyperbolic manifolds and F : M → N be a holo-morphic isometry.

(a) If p, q ∈ M and [p, q] is a geodesic segment joining points p, q of M thenF ([p, q]) is a geodesic segment joining F (p), F (q) in N.

(b) If c : (−∞,+∞) → M defines a geodesic of M, then c′ = F c defines ageodesic of N.

Proof. It suffices to prove a). Suppose that c defines the geodesic segment [p, q] ofM and let t1 ≤ t2 ≤ t3 ∈ [a, b]. Then, F ([p, q]) is defined by c′ = F c and

κN

(c′(t1), c

′(t3))

= κN

((F c)(t1), (F c)(t3)

)= κ

M

(c(t1), c(t3)

)= κ

M

(c(t1), c(t2)

)+ κ

M

(c(t2), c(t3)

)= κ

N

((F c)(t1), (F c)(t2)

)+ κ

N

((F c)(t2), (F c)(t3)

)= κ

N

(c′(t1), c

′(t2))

+ κN

(c′(t2), c

′(t3)).

Uniqueness of geodesics in a hyperbolic manifold is not in general ensured byany condition concerning the metric. For instance, completeness of the metric maynot be enough.

For reasons of clarification, we shall describe below a standard failure of unique-ness in the case of the hyperbolic polydisc (∆, d e∆). It is well known that there is aunique geodesic segment joining any two points of the polydisc when the latter isequipped with the Bergman metric. Unfortunately, the picture in our case is notthat good.

Proposition 3.7. Let (∆, d e∆) be the hyperbolic polydisc. Then for each two distinctz, w ∈ ∆, z = (z1 , ..., zn

), w = (w1 , ..., wn), there exists a unique geodesic segment

joining them only if and only if all distances d∆(zi, w

i) are equal. Otherwise there

exist infinitely many geodesic segments joining these two points.

Proof. The result follows immediately from the proof of Proposition 3.6. We onlynote that in the case where all distances d∆(z

i, w

i) are equal, then the hyperbolic

metric coincides with the Bergman metric and the induced geodesic segment isunique in the standard sense.

Notice also that apart from geodesic segments joining two points in the polydiscwhich are obtained by holomorphic isometries, there also exist others which are notof this kind. We shall give an example in the case n = 2 where we can find atleast two such segments. The modifications needed for the general case will thenbe apparent.

We suppose that z = (0, 0) = 0 and w = (a1 , a2) = a, ai ∈ (0, 1). We shall treatthe case when a1 < a2 and a2 − a1 > a1 . All other cases may be carried out in thesame manner.. We consider the euclidean segments

[(0, 0), (0, a1)]e ∪ [(0, a1), (a1 , a2)]e


and[(0, 0), (a1 , a1)]e ∪ [(a1 , a1), (a1 , a2)]e

joining (0, 0) and (a1, a2). Straightforward calculations show that these segmentsare also geodesic with respect to the hyperbolic metric and can not be the imagesof any f ∈ I(∆, ∆).

More generally speaking, given a hyperbolic manifold M with Kobayashi metricκ

M, then to each two distinct points p, q of M we may assign a subset G(p, q) of M

defined byG(p, q) = r ∈M ;κ

M(p, r) + κ

M(r, q) = κ

M(p, q).

G(p, q) enjoys several properties, a number of which we state below. Their proof isstraightforward.

(1) G(p, q) is a closed subset of M and is always different from the null set sinceit contains at least p, q.

(2) M =⋃

p,q∈M,p 6=q G(p, q).

(3) Every geodesic segment joining p and q lies entirely in G(p, q), and from eachpoint of G(p, q) passes at least one geodesic segment.

(4) diam(G(p, q)) = maxκM

(r1 , r2); r1 , r2 ∈ G(p, q) = κM

(p, q).

(5) If F ∈ I(M) is such that F (p) = p′ and F (q) = q′, then F (G(p, q)) = G(p′, q′).It is therefore natural to ask if among all segments in G(p, q), there exists a

unique one characterised by a special property. We have seen in Proposition 3.4that such a segment actually exists in the case of the polydisc.

In the rest of this section we study more general hyperbolic manifolds in whichthis situation also appears.

Definition 3.8. A hyperbolic manifold M with hyperbolic metric κM

is called anI-manifold if the following condition holds.

(A) For every two different points p, q in M there is a holomorphic isometry f ∈I(∆,M) such that f(0) = p and f(a) = q where a = tanhκ

M(p, q).

It follows from the definition that any two points of an I-manifold can be joinedby at least one geodesic segment. For our purposes we shall also need

Definition 3.9. An I-manifold M is called an IU-manifold if

(a) for every two different points p, q in M there is a unique holomorphic isometryf ∈ I(∆,M) such that f(0) = p and f(a) = q where a = tanhκ

M(p, q) and

(b) if there is an h ∈ H(∆,M) such that h(0) = p, h(a) = q and for each t ∈ (0, a),

h(t) ∈ [p, q] = [f(0), f(a)],

then h ≡ f.

Note. An interesting question arising from the definition is for which manifoldscondition b) is always satisfied. It is easy to see that condition b) implies h = fon [0, a]. Indeed for each t ∈ (0, a) there is an s in (0, t] such that h(t) = f(s). (smight not be in (t, a): by the distance decreasing property κ

M

(p, h(t)

)≤ d∆(0, t)).


Now,

d∆(0, a) = κM

(p, h(t)

)+ κ

M

(h(t), q

)≤ d∆(0, s) + d∆(t, a)

and sinced∆(0, a) = d∆(0, t) + d∆(t, a)

we haved∆(0, t) ≤ d∆(0, s).

Since s ≤ t only equality can hold true, and thus s = t and h(t) = f(t).Following a standard analytic continuation argument one can prove that whenM

is (equivalent to) a bounded domain in Cn (or more generally in an n dimensionalcomplex Banach space) then h = g everywhere in the disc.

The next propositions will be needed in the next section.

Proposition 3.10. Let M be a cross product of I-manifolds Mi, i = 1, ..., n with

hyperbolic metric κfM = maxκM

i. Then M is an I-manifold.

Proof. Let p = (p1 , ..., pn), q = (q1 , ..., qn) be points in M and ai = tanhκM

i(pi , qi),

a = tanhκfM (p, q). Assume for simplicity that all ai are different from 0. Since allMi are I-manifolds, there exist f

i∈ I(∆,M

i) such that f

i(0) = p

iand f

i(a

i) = q

i.

We distinguish two cases:(a) a1 = ... = a

n= a. Then f = (f1 , ..., fn

) belongs to I(∆, M) as this followsfrom Proposition 3.7 (a) and clearly f(0) = p and f(a) = q.

(b) ai 6= aj for at least one pair of distinct i, j. Define F : ∆ → M by

F (z1 , ..., zn) = (f1(z1), ..., fn

(zn)).

Then from Proposition 3.7 (b) we have that F ∈ I(∆, M), F (0, ..., 0) = p andF (a1 , ..., an

) = q. Let g : ∆ → ∆ given by

g(z) =(a1

az,a2

az, ...,

an−1

az,a

n

az)

be the holomorphic isometry of the disc into the polydisc as in the proof ofProposition 3.6. Here a = maxa

i; i = 1, ..., n. Let f = F g. Then,

fi(z) = f

i

(ai

az), f(0) = F

(g(0)

)= F (0, ..., 0) = p,

f(a) = F(g(a)

)= F (a1 , ..., an) = q

and f ∈ I(∆, M). Thus M is a I-manifold.

Note. The isometry f = (f1 , ..., fn) constructed above satisfies an additional prop-

erty


(B) for each i = 1, ..., n fi([0, a] = [f

i(0), f

i(a

i)] where the latter is the geodesic

segment joining pi

and qi

in Mi

obtained by fi.

Indeed, if t ∈ (0, a], fi(t) = f

i(a

i

a t) ∈ [fi(0), f

i(a

i)] since a

i

a t ≤ ai. Conversely,

if r ∈ (fi(0), f

i(a

i)], there exists a t ∈ (0, a

i] such that r = f

i(t) = f

i( a

ait) and

aa

it ≤ a.

The following generalises Proposition 3.6.

Proposition 3.11. Let M be a cross product of IU-manifolds Mi, i = 1, ..., n

with hyperbolic metric κfM = maxκM

i. Then for each two points p = (p1 , ..., pn

),

q = (q1 , ..., qn) of M, there exists a unique element f ∈ I(∆, M), f = (f1 , ..., fn

)such that:(A) f(0) =p and f(a) = q where a = tanhκfM (p, q).

(B) For each i and for each t ∈ (0, a],

fi(t) ∈ [f

i(0), f

i(a

i)]

where [fi(0), f

i(a

i)] is the geodesic segment joining p

iand q

iin M

idetermined

by the unique fi∈ I(∆,M

i) which are such that f

i(0) = p

iand f

i(a

i) = q

i,

ai = tanhκM

i(pi , qi).

(C) For each i such that ai< a the quantities κ

Mi(p

i, f

i(a

i)) are maximal in the

sense that if there exists a h = (h1 , ..., hn) in I(∆, M) different from f andsatisfying conditions (A) and (B) then

κM

i

(pi , fi(ai)

)> κ

Mi

(pi , hi(ai)

)unless hi(ai) = fi(ai).

Proof. We construct f as in Proposition 3.10 and distinguish two cases:(a) a1 = ... = a

n. Then,

f = (f1 , ..., fn)

and we shall show that f is the unique element of I(∆, M) which satisfies (A)and (B).Indeed, if h = (h1 , ..., hn

) ∈ I(∆, M) satisfies (A) and (B), then since hi∈

H(∆,Mi), hi(0) = pi hi(a) = qi and hi(t) ∈ [fi(0), fi(ai)] for all i, we obtainthat h

i≡ f

ias this follows from Definition 4 b).

(b) ai6= a

jfor at least one pair of distinct i, j. For each i,

fi(z) = fi

((ai/a)z

).

Let again h = (h1 , ..., hn) ∈ I(∆, M) different from f and such that it satisfies

(A) and (B). Suppose first that some hj∈ I(∆,M

j). Then by (B), a

i= a and

thus hj

is identical to fj which in this case is identical to fi. So let us suppose

that there is some hi/∈ I(∆,M

i) and therefore a

i< a. Then if t ∈ (0, a

i],

κM

i

(p

i, h

i(t))≤ d∆(0, t) = κ

Mi

(p

i, f

i(t)).


For t = ai

we then have

κM

i(pi , hi(ai)) ≤ κ

Mi(pi , fi

(ai))

< κM

i

(pi , fi(a)

)= κ

Mi

(pi , fi(ai)

)and the proof is complete.

Definition 3.12. If f satisfies the conditions of Proposition 3.11 then it is calledthe maximal geodesic isometry of ∆ into M (assigned to the points p, q).

The geodesic segment induced by f is called the maximal geodesic segment join-ing p, q and the geodesic f

((−1, 1)

)the maximal geodesic passing through p and

q.

Proposition 3.4 may now restated as follows:

Proposition 3.13. Let (∆, d e∆) be the hyperbolic polydisc. Then for each z, w ∈ ∆z = (z1 , ..., zn

), w = (w1 , ..., wn) there exists a maximal geodesic segment joining

them. This segment is identical to the one obtained when ∆ is equipped with thestandard Bergman metric.

4. Application to T , T(G) and QC (G).

We start with the following theorems which can be found in [4] (Theorems 5 and6 respectively):

Theorem 4.1. Let Γ be a Fuchsian group, f ∈ H(∆, T eich(Γ)) and t0 ∈ ∆.Suppose that

τT (Γ)

(f(t0), f(t)

)= d∆(t0 , t)

for some t ∈ ∆. Then f ∈ I(∆, T eich(Γ)).

Theorem 4.2. Suppose that 0 6= µ ∈ Belt(Γ) is extremal. Then the following areequivalent:(a) µ is uniquely extremal and ‖ µ ‖∞=| µ | almost everywhere,

(b) there is only one geodesic segment joining 0 and Φ(µ),

(c) there is only one f ∈ I(∆, T eich(Γ)) such that f(0) = 0 and f(‖ µ ‖∞) = Φ(µ),and

(d) there is only one g ∈ H(∆, Belt(Γ)

)such that g(0) = 0 and Φ

(g(‖ µ ‖∞)

)=

Φ(µ).

Theorem 4.3. Let Teich(Γ) be the Teichmuller space of a Fuchsian group Γ andτ

T (Γ) be the Teichmuller-Kobayashi metric on Teich(Γ). Given two distinct pointsφ, ψ in Teich(Γ) there exists a geodesic segment joining them. If Teich(Γ) is finitedimensional or if each non-zero extremal µ ∈ Belt(Γ) is uniquely extremal with‖ µ ‖∞=| µ | almost everywhere, then this segment is unique.


Proof. (Outline) It suffices to assume that ψ = 0. By Teichmuller’s ExistenceTheorem there exists an extremal Beltrami differential 0 6= µ ∈ Belt(Γ), such thatΦ(µ) = φ. The image of f : [0, ‖ µ ‖∞ ] → Teich(Γ) where for each t in [0, ‖ µ ‖∞ ],

f(t) = Φ(tµ/ ‖ µ ‖∞)

is a geodesic segment joining 0 and φ. We can extend f in the obvious way toa holomorphic mapping of the unit disc into Teich(Γ) and then from Theorem2.10 we have that f ∈ I(∆, T eich(Γ)). If Teich(Γ) is finite dimensional then byTeichmuller’s Uniqueness Theorem each extremal Beltrami differential µ ∈ Belt(Γ)is uniquely extremal and is of the form

µ = λ| ϕ |ϕ

where λ ∈ [0, 1) and ϕ ∈ Q(Γi) i.e. it is a Teichmuller differential. Therefore

condition (a) of Theorem 2.11 is satisfied and we obtain the result. In the infinitedimensional case, condition a) is satisfied by assumption.

Let f be as in the proof of Theorem 2.14. The f -image of the unit disc iscalled a Teichmuller disc centred at φ. Additionally, there is an isometry of Ronto f((−1, 1)) and therefore every Teichmuller geodesic is isometric to the realEuclidean line. Therefore Teich(Γ) is a straight space.

Theorem 2.14 can be restated as below.

Theorem 4.4. Let Teich(Γ) be the Teichmuller space of a Fuchsian group Γ andτ

T (Γ) be the Teichmuller-Kobayashi metric on Teich(Γ). Then Teich(Γ) is an I-manifold. If Teich(Γ) is finite dimensional or infinite dimensional and such thateach non-zero extremal µ ∈ Belt(Γ) is uniquely extremal with ‖ µ ‖∞=| µ | almosteverywhere, then it is an IU-manifold.

Proof. We only have to check condition b) of Definition 4. In the finite dimensionalcase this is automatically satisfied as this follows from the note following Definition4. But in general, we can work as follows.

Let as before 0 and φ points of Teich(Γ) and suppose that φ = Φ(µ) where µ isuniquely extremal and ‖ µ ‖∞=| µ | almost everywhere. Suppose that there existsan h ∈ H(∆, T eich(Γ)) such that h(0) = 0 and h(‖ µ ‖∞) = φ = Φ(µ). FromLifting Theorem in §2, there exists a g ∈ H(∆, Belt(Γ)) such that Φ g = h andwe can choose it so that g(0) = 0. Then g also satisfies

(Φ g)(‖ µ ‖∞) = h(‖ µ ‖∞) = φ = Φ(µ)

and thus it is the uniquely defined mapping of Theorem 9 d). Again, LiftingTheorem induces that the mapping f in the proof of Theorem 10 is lifted to thesame g and Φ g = f. Thus f = h and our assertion is proved.

We now state the main theorem of this section:

Theorem 4.5. A cross product T of Teichmuller spaces Teich(Γi) of Fuchsian

groups Γi

is an I-manifold: Given two distinct points φ = (φ1 , ..., φn) and ψ =

(ψ1 , ..., ψn) of T there is an f ∈ I(∆, T ) such that f(0) = φ and f(a) = ψ where

a = τ eT (φ, ψ) = maxi=1,...,n

ai= τ

T (Γi)(φi

, ψi).


Therefore [φ, ψ] = f([0, a]) is a geodesic segment joining them.If T is finite dimensional or any extremal Beltrami differential µi of Belt(Γi)

is uniquely extremal and satisfies ‖ µi ‖∞=| µi |almost everywhere, then there is amaximal geodesic segment joining φ and ψ.

Proof. For each i, Teich(Γi) is an I-manifold as this follows from the proof ofTheorem 10. From Proposition 5 we also deduce that T is also an I-manifold.Now if each Teich(Γ

i) is finite dimensional, or each extremal Beltrami differential

is uniquely extremal and satisfies the condition of the theorem, then again fromTheorem 10 it is an IU-manifold. Our last assertion is thus obtained by Proposition6.

Corollary 4.6. Let T (G) (resp. QC(G)) be the Teichmuller (resp. the Quasicon-formal Deformation) space of a finitely generated Kleinian group with hyperbolicmetric κbT (resp. κQ). For any two distinct points, there is a maximal geodesicsegment joining them.

We now wish to describe the form of the maximal geodesic isometry f whenψ = 0 and φ = (φ1 , ..., φn) and dim(T ) < +∞. From the general construction of fin Proposition 3.11 and the proof of Theorem 2.14, we obtain that

f(z) = Φ(

µ1

‖ µ ‖∞z, ...,

µn

‖ µ ‖∞z

)where Φ = (Φ1 , ..., Φn

), µi

are the unique extremal differentials such that Φi(µ

i) =

φi

and‖ µ ‖∞ = max

i=1,...,n

‖ µi ‖∞.

Thus the f -image of ∆ is an n dimensional complex manifold which is a crossproduct of Teichmuller discs centred at φi and possibly single points. We call thisset a Teichmuller n−polydisc centred at φ.

Finally, T has a straight space property. Let f((−1, 1)) be the maximal geodesicof T passing throughout the origin and φ. It is straightforward to prove that themapping

R → f((−1, 1))such that

x → f(tanhx)

for every x ∈ R is an isometry of the real Euclidean line into T .

5. A Note on Moduli Spaces

Let as usual T (G) be the Teichmuller space of a Kleinian groupG and Γ1 , ...,Γnbe the Fuchsian model for G. Assume further that the signature (gi , ni) of Γi isso that 2g

i+ n

i> 4 for each i = 1, ..., n. Following Gentilesco, we call such a G

non-exceptional.As it follows from Theorem 4.2, the Modular group

M(G) =n∏

i=1

Mod(Γi)


acts properly discontinuously on T (G) as a subset of hyperbolic isometries. Definenow the Moduli space of G to be

R(G) = T (G)/M(G).

It is clear that

R(G) =n∏

i=1

Riem(Γi)

where Riem(Γi) = Teich(Γ

i)/Mod(Γ

i) is the classical Moduli space of Γ

i. This space

is an orbifold and an irreducible normal complex space. The Teichmuller-Kobayashimetric κeT projects to a complete hyperbolic metric in R(G) by virtue of Proposition6.3 in [9]. It has been shown by C. McMullen in [12] that Riem(Γ) admits acomplete Kahler metric g comparable to the Teichmuller-hyperbolic metric, withrespect to which it has finite volume.

Proposition 5.1. The Moduli space R(G) of a non exceptional Kleinian groupG admits a complete Kahler metric g comparable to the hyperbolic metric. Withrespect to this metric, R(G) is of finite volume volg . Explicitly, if vol

gi

is the finitevolume of the Moduli space of each Γi , then

volg = volg1× ...× vol

gn.

Proof. Just set g = g1 × ... × gn that is the product Riemannian metric. Wefirstly prove that g is comparable to the hyperbolic metric κ eR of R(G). To do so, itsuffices to prove comparability in the universal covers. Let g = g1 × ...× gn

be themetric in T (G). Since gi is comparable to τ

T (Γi) for each i, we easily obtain that g

is comparable to τ eT (G). Now g is complete and Kahler, and by Fubini’s Theorem

we obtain the desired result.

References

1. L. Bers. Spaces of Kleinian groups, Lecture Notes on Mathematics, Vol. 155,Springer-Verlag, 1970, pp. 9–34.

2. C.J. Earle and F.P. Gardiner. Geometric automorphisms between infinite di-mensional Teichmuller spaces, Trans. Am. Math. Soc. 348 (no. 3) (1996),1163–1190.

3. C.J. Earle and I. Kra. On Holomorphic Mappings Between Teichmuller Spaces,Contributions to analysis, Academic Press, New York, 1974.

4. C.J. Earle, I. Kra, and S.L. Krushkal, Holomorphic motions and Teichmullerspaces, Trans. Am. Math. Soc. 343 (1994), 927–948.

5. F.P. Gardiner, Teichmuller Theory and Quadratic Differentials, Wiley-Interscience, New York, 1987.

6. J.A. Gentilesco, Automorphisms of a deformation space of a Kleinian group,Trans. Am. Math. Soc. 248 (1979), 207–220.

7. M. Jarnicki and P. Pflug, Invariant distances and metrics in complex analysis,De Gruyter Expositions in Mathematics 9, W. de Gruyter and Co. Berlin,1993.

8. I. Kra. On spaces of Kleinian groups, Comment. Math. Helv. 47 (1972), 53–69.9. S. Kobayashi. Hyperbolic Manifolds and Holomorphic Mappings, Marcel-

Dekker, New York, 1970.


10. O. Lehto. Univalent Functions and Teichmuller Spaces, Springer-Verlag, NewYork, 1987.

11. Li Zhong, Non uniqueness of geodesics in infinite dimensional Teichmullerspaces, Complex Variables Theory Appl. 16 (1991), 261–272.

12. C.T. McMullen, The moduli space of Riemann surfaces is Kahler hyperbolic,Ann. of Math. (2) 151 no. 1 (2000), 327–357.

13. K. Matsuzaki and M. Taniguchi. Hyperbolic Manifolds and Kleinian Groups,Clarendon Press, Oxford, 1998.

14. R. Narasimhan, Several Complex Variables, Chicago University Press,Chicago, 1971.

15. H.L. Royden, Automorphisms and Isometries of Teichmuller space, Advancesin theory of Riemann surfaces, Ann. of Math. Stud. Vol. 66, Univ. PrincetonPress, NJ, 1971, pp. 369–383.

Dr Ioannis D. PlatisDepartment of Mathematical Sciences

University of Durham

Durham DH1 3LE

UNITED KINGDOM

Current Address:

29 Rafkou Street GR 71304 Neo StadioHeraklion Crete

Greece

EUROPE

johnny–[email protected]

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