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### 1 (1993), p. 79-98

<http://www.numdam.org/item?id=CM_1993__87_1_79_0>

© Foundation Compositio Mathematica, 1993, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

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(2)

on

### the space of space forms*

JILL McGOWAN

Howard University, Washington, D.C.

Received 21 May 1992; accepted 2 July 1992

(0 1993 Kluwer Academic Publishers. Printed in the Netherlands.

0. Introduction

A manifold whose fundamental group is finite and whose sectional curvature is bounded cannot have an

### arbitrarily

small diameter. This

for

### example,

from Gromov’s result that for every dimension n, there exists

### 8(n)

&#x3E; 0 such that if the sectional curvature

### kM

of an n-dimensional manifold M is

and the diameter

of M satisfies

M is

covered

a

nilmanifold

In

### particular,

the fundamental group

### 03C01(M)

is infinite. Since a

manifold of

### positive

Ricci curvature has finite fundamental group

### ,

such a

manifold cannot have a diameter less than

A

space

that

### is,

a

manifold for which

=

### 1,

must therefore have a diameter

than or

to Gromov’s

Gromov’s

### result,

of course, has far more

than

### simply forcing

a lower bound on the diameter of space forms.

other

### things,

it

forces a lower bound on the diameter of any manifold with bounded curvature and finite fundamental group. In

### spite

of the wide range of manifolds to which these lower diameter bounds

### apply,

the values of those lower bounds were not

even in the

### simplest

case - the case of manifolds of constant

### positive

curvature. In this paper, we determine lower bounds on the diameters of

### spherical

spaces forms and prove the

theorem:

THEOREM 0.1. The

diameters

space

is

arc-

not,

actual

any

10

is not,

### however,

the actual diameter

any space

This is an

lower

### bound,

since it is the actual diameter of a space in the Hausdorff closure of space forms. It occurs in dimension three. Dimension three is

### special

for several reasons; not

### only

does the minimal lower bound on the diameter of space forms occur with

### respect

to three-dimensional space

### forms,

but the groups that act on

to

### produce

these space forms are the

### building

blocks for groups that

### produce

space forms of all other dimensions. These groups have been

described

J. Wolf.

### Except

for reducible actions

*This paper contains the results of the author’s Ph.D. dissertation, written under the supervision of

Karsten Grove of the University of Maryland.

(3)

and two other

### types

of groups, all the elements of these groups may be

as block

### matrices,

in which each block is a

### two-by-two complex

matrix that appears as an element of a fundamental group of a three- dimensional space form. In each such group, at least one

the

### complex

coordinates in blocks of two. These groups

occur as

of

and act on

In

the group

### permutes

the coordinate three-

in

and acts on each

### three-sphere

as one of the groups

### acting on S3.

The lower bounds we have determined for space forms are

on

### dimension,

and increase as dimension increases. In every case, these lower bounds increase to

### 03C0/2

as the dimension increases without bound.

a

### complete

list of the lower bounds on space

see p.

In the even

the

### only spherical

space forms are the

and

### projective

space. The smallest diameter that occurs when the curvature is normalized to one is therefore

As a consequence

this lower bound

also for

### positively

curved manifolds whose curvature is bounded above

one,

### not just

to those of constant

### positive

curvature. Since the even-dimensional lower bounds

### apply

to manifolds of

variable curvature, we are

with the

### question:

Does the lower bound

determined here

### apply

to manifolds of variable curvature also?

1. Limit spaces of space forms

### Joseph

Wolf has classified

### spherical

space forms. In each odd

there

are

many

are of the form

### Sn/0393,

where r is a finite

of

### SO(n + 1).

Wolf described these space forms

the

### underlying

groups r. He enumerated six

### types

of groups. While each group is

groups of each

### type

may be found that are

a

### complete

list of the groups, see p.

### 16.)

Let K be the set of

### compact

subsets of S". If G is a closed

of

### SO(n + 1),

the orbits Gx in sn are

### compact

and therefore are elements of K. We define distance on K as the Hausdorff

=

&#x3E; 0: the e-

### neighborhood

of X contains Y and the

of Y contains

Under

this Hausdorff

K is a

set

### .

Observe that we may consider the space forms

as subsets of K.

on this

we need not

### expand

the definition of Hausdorff distance

the

one

### given

above.

We would like to determine when a sequence of these space

viewed as a

subset of

### K,

converges in the Hausdorff sense. We

on the

easy result:

THEOREM l.l.

close to

in the

in

+

is

close to

in the

sense in K.

(4)

If

### ri

converges to a set G in

+

G is a

of

+

From

Theorem

### 1.1,

we can conclude that if

### Fi

converges to G in the Hausdorff sense,

converges to

### S"IG

in the Hausdorff sense.

if M is a limit

in

the set

then M =

where

### F;

is an infinite sequence of groups at a

enumerated

Wolf. This fact

the easier

### part

of our second theorem:

THEOREM 1.2.

M is in the

closure

the set

then

M =

where G is a

+

whose

### identity

component is a q- dimensional torus Tq and whose

the

component

or else G

group.

We use the

### following

theorem of C. Jordan: If S is a connected Lie group, then there exists an

r =

with the

### property

that if F c S is a finite

of

### S,

then F admits an abelian normal

such that

In our

### case, S is SO(n + 1 ).

We have a sequence of finite

c S such

that

### Fn

converges to a group G c S.

the theorem of

for each

there

exists

### F"o,

an abelian normal

in

We consider

= lim

We

claim that

### Go

is an abelian normal

of G and that

### Go

is abelian and normal in G because its elements are limits of elements in the

and because

is continuous in

If

### Go

has more than r cosets in

### G,

then we have at least r + 1 distinct sequences

such that

Since

for i

there is a

distance between

and

is because

is

But then for n

So

### F no

has at least r + 1 distinct cosets. This contradicts the theorem. Therefore

### [G : Go]

r, so dim G = dim

Since I

the

of

### G, G°,

must be contained in the

normal

### subgroup Go.

We can conclude that

is abelian.

Since

is a

### compact, connected,

abelian Lie group, it is

to

the

### q-dimensional

torus, for some q ~N.

is

because G is

### compact,

and the order of G is the number of connected

of G. This

the

### Often,

it is easier to describe the space

than

### X/G.

We therefore will

make

### repeated

use of the fact that

=

### X/Go

Another fact that we will find useful is stated in the

lemma:

LEMMA 1.3

is a closed

a closed group

isometries G on

### M, M/H

has a diameter at least as

as

### M/G.

In this paper, we will consider

odd-dimensional

S" and

### only

irreducible actions

the groups

### r,

since the other results are well-known.

(5)

2. Three-dimensional space forms

The

### spherical

space forms of dimension three are of the form

where

r c

### SO(4)

is a finite group that acts

and

on

### S3.

We recall from Wolf that the

### possible

irreducible groups r may be described as

follows

### .

Case 1. The group is

These

are

### subject

to the further restrictions that t = 0 mod

=

= 1 and

1 mod m.

Case 2. r is

the two

to the same

### generator:

either

L J

where v - 0 mod 2 and

= 1.

Cases 4-6. ris

x

x 0* or

### 7Lu

x I*. These groups,

### T*,

0* and I* are the

of the

### tetrahedral, octahedral,

and icosahedral groups in the unit

### quaternious

under the cover map

x

= Ax for

all

the

### product q x q-1,

x is treated as a

that

### is, (x1, x2, x3)

is identified with

and

are

### augmentations

of T* and 0*. That

is formed

a

### generator

of order 3v for the

### generator

of order 3 in T*.

THEOREM 2.1. Let

group

### of a type

described in cases

2 and 3. Then

the diameter

bounded

=

an

interval.

### Proof.

We prove the theorem for case 1 groups. The other

### proofs

are similar.

We note that the extra

### generators

in case 2 are

as elements of the limit group in case 1. We observe that

is a scalar

### matrix, B2 = e41til/t.

1. As m and t become

and

converge to

(6)

circle

since

converges to

1 : 03B8 ~

and

converges to

(p E

c ~

### 1 . Therefore, ~A, B2~

converges to a two-torus of the form

and 03B8 E

The orbit space

### S3/T2

is

therefore one dimensional. Since we can write

as x =

r2

where

we find that the

for

= 1 or ri = 0

rise to

### isotropy.

This tells us that

### S3/T2

is an interval and

= 1

= 0

to the

of that interval

The diameter of

### S3/T2

is therefore the distance from

to

in the

it is

If G =

### generator

outside of this two-

torus. It is the limit of B as t ~ oo :

is

the connected

of the

in G. So

Since

This last

and

the

fixed. In

### effect,

it folds the interval in half. The diameter of

is

THEOREM 2.2.

r =

x

### Tg, S3/0393

has diameter at least

I-’ =

x

### 0*, S3/0393

has diameter at least

arccos

0393 =

x

has diameter at

arccos

### 1 tan 37r

In each case, the limit is topo-

### Proof.

Now we consider space forms of the

where r is the

of

for some

and some

a

0* or 1*.

For any such

### représentation, ~ e203C0i/u is a subgroup.

As u becomcs very

For any such

is a

### subgroup.

As u becomes very

### large,

this converges to the

action on

~

### (eiezl eiOz2)’

The limit

space is therefore

diameter

The finite group

### (G/S1)

that acts on this

is then

or

That

since -I E

E

### [0,203C0)},

the actions of A and - A in

### T*, 0*,

and I* are identified on

These are the

actions

that are

since

=

for

if for any

### z ~ S3

there exists some 0 ~

such that

=

But then

=

so z is an

for all

and

is an

The

elements

0* and

I* with this

are

### + I,

since no other elements of the groups have two

and

### S3/S1XT*=S2/T. Therefore,

the

diameter of the limit space is the diameter of

is the

of

### S2/T

has fundamental domain shown in

### Fig.

1. Under the action of the tetrahedral group, AB is identified with

### AC,

and BD with DC. The

limit

space is

but not a

### symmetric sphere.

Its diameter is the

(7)

Fig. 1.

### length

of the curve AB. In the

of

this is

in

### S2/0,

we have the fundamental domain on the

to the

2.

AB and AC are

### identified,

as are BD and CD.

The diameter of this

### triangular "envelope"

is the distance AC. In the

of

### diameter 03C0/2,

the distance AC is

or

## 1 arccos 1. Similarly,

the

fundamental domain in

under the action

1

in

3. AB and AC are

### identified,

as are BD and CD. The diameter of

therefore

to the distance

which in the

of

is

We have omitted

### discussing

the difference between

and T*. T* has

=

and P and

where P and

to

X =

P and

where P and

to

across two

### edges

of the tetrahedron in the tetrahedral group.

has

(8)

Fig. 2.

Fig. 3.

(9)

and

where

In other

### words,

the substitution of

### Xv

for X does not affect the

case.

3.

### Space

forms of dimension n = 1 mod 4

In this

### section,

we will prove the

### following

theorem:

THEOREM 3.1. The diameter

a space

where n _ 1 mod

is

bounded below

arcsin

bounded below

arcsin

### n-1

We find this limit

### by exploring

the action of one

of group on the

S". As

### previously noted,

Wolf enumerated six

### types of groups

in the classification of space forms.

### (The

list of the groups is located in Section

one

of group acts on

### S",

where n ~ 1 mod 4: groups of

I.

a theorem of

### Vincent,

a group r that is the fundamental group of a space form of dimension n ~ 1 mod 4 is a group with two

A and

### B,

and relations Am = Bt =1 and

### BAB-1= Ar .

The order of r satisfies

= mt =

in

=

### 1,

and where d is the order of r in the

### multiplicative

group of units in

### 7Lm.

Wolf showed that under any irreducible

### representation

03C0k,l of a group of

the

### images

of A and B are as follows:

and

(10)

where

### (k, m) = 1 = (l, t’)  .

A and B are d x d

so

these are the

### only

irreducible actions on

### S",

where n - 1 mod 4. We note that

### Bd

is a scalar matrix:

### (e203C0i/t’).

I. As m and t become

the

### subgroup generated by 03C0k,l(A) and 03C0k,l (Bd)

converges to a torus action on the

the torus

### having

dimension at least two, since

and

In

### fact,

as m increases without

### bound,

the condition that d be the order of r in the

### multiplicative

group of units in

### Zm

can result in the

close to any

in the d-1 torus

The limit group

these

will never be

a

### d-torus,

since we have the relation

The

### following

theorem is due to Lawrence

### Washington.

THEOREM 3.2. For some sequences

r and m,

### ~03C0c,l(A) ~

converges to a d -1

torus as m increases without bound.

the

see

of whether

### 03C0k,l(A))

can converge to a smaller

dimensional torus

a

### suitably

chosen sequence of r’s and m’s.

We also note that

converges to an

### independent

circle action. In the

### limit, then, 03C0k,l(A), 03C0k,l(Bd)~

can converge to a d-torus.

LEMMA 3.3. The diameter

the limit space

where G = lim

is bounded

below

### by

arcsin where 2d = n + 1.

### Proof: Regardless

of the dimension of the torus

on

### S", however,

the distance between the orbits of

and remains the

same.

no

of direction

### (eiO, 0, ... , 0) brings

the orbits any closer

than when

nor does any

the

affect their

since the

unaffected

this

(11)

So with

### respect

to any limit space

where

k =

or

we have the

### following

lower bound on diamete

LEMMA 3.4. The diameter

any

is bounded below

^

v --

### Proof. Any

finite group r will be contained in a group

With

to the elements

and

### 03C0k,l(Bd),

the containment is

since these are

### diagonal

matrices whose entries are of the form

for some

For any

7rk,l, the

is a

The left

is an element of

the

is included

as a

of G. Since the

### generators

of every r = Im nk,l are contained in

### G,

r c G. Since we have containment of the groups

### r c G, by

the inclusion lemma

the diameter of

is bounded below

that of

(12)

the

of

1 groups are the

### only

irreducible actions on the

### spheres

S" with n ~ 1 mod

### 4,

these groups also serve as fundamental groups of

### spheres

of other dimensions. All the same conclusions reached in this section with

### respect

to the limits of groups

### of type

1 and the lower bound on diameter of the

space forms

### apply

in all dimensions.

4.

induced from

on

### sphères

of dimension 3 mod 4

Within the six

### types

of fundamental groups of

space

### forms,

Wolf

defined several cases.

### Using Fc(G)

to denote the set of

classes of

free

of a group

### G,

Wolf describes 13

classes

from which all

### representations

of fundamental groups of space forms are derived. The classes

and

in the

### following

table.

For the purposes of this paper, it shall be convenient to deal with the

ak,l and

as a

and

from the

of

groups of

### types III, IV, V,

and VI. These latter groups may be

to act on

as will be

### explained

further in the next section. It is less convenient to think of (Xk,l in this way, and not useful to

in this

fashion.

(Xk,l and

### Pk,l

are both built on the

1tk,l’

Table 1

(13)

### Representations

ak,l are induced from nk,l.

"induced

we

mean the

cr

in block form

### by ij g)= a(bi-lgbj),

where u is

defined on a normal

the

constitute a

set of coset

of that

### subgroup

in the whole group, and

is taken to

be zero when

is not in the

Under this

akn

where the

is

### A, B&#x3E;.

Since the whole group is

and R

there are

two cosets:

and

### RA, B&#x3E;.

So

Note that in each row and in each

there is

one nonzero

If

g E

then

### Rg ft A, B&#x3E;,

and so on.

THEOREM 4.1. Arcsin

### n n + 1 1 is

a lower bound on the diameters

n+1

the type

where r is the

a

### finite

group under ak,l or

### Proof.

We have seen that for all k and

1 - ....II 1

with

R. We have

and

if r =

(14)

When ak,j is

Otk,l =

O

a

of half the

if r =

r c

=

where

is a

of the

group on d

matrices which

coordinates.

The orbit of

under

is

es : 0 e

### [0, 203C0),

es is the s th standard normal coordinate vector, SE

### {1, 2,..., 2d}}.

This is because the torus action

serves to

the

vector

and the

### permutation subgroup,

is a transitive action on the coordinate axes. The distance between a

and the orbit of

... ,

is

arcsin

2d Here n + 1=

so

arcsin

## 2d -1= arcsin n 1.

2d n + 1 Therefore, diameter

### Sn/G03B1 l nr- 1

n + 1

Since all the finite groups rare contained in

the diameter of

is

bounded

this number

### by

the inclusion lemma.

The

for

follows

the same

that if

r has

so the

### diagonal

element is

and the distance to the orbit of

is

arcsin

5. Actions

groups of

V and VI on

### sphères

of

dimension 3 mod 4

Our

for

### determining

lower bounds on the diameters

### of space

forms with

fundamental groups of

### types III, IV,

V and VI is to consider the action of the

(15)

on

### pairs

of coordinate axes. That

any

written

as a

in

### C2n+2,

has an even number of coordinates. If all but one

of

coordinates

### (say (zl, Z2))

are zero, the nonzero

### pair

of coordinates may be

of as a

in

since then

=1.

### Pairing

the coordinates this way, we think of each

as a coordinate

in

### S4n + 3.

The actions under groups of

III

### through

VI treat these coordinates in

That

the

### subgroups

of the torus actions on these

are

scalar

### multiplica-

tion on each coordinate

converge to the

### Hopf

action on each coordinate

in the limit.

the

### permutation

elements of the groups

the

coordinates in

are no

### longer

transitive actions on the

coordinate axes

but

on the

### pairs

of coordinate axes. For

### replaces

as an element of the

### permutation subgroup

in the limit groups.

In

### fact,

as we shall see, the groups all converge to a group of the form

### (Td, P(d), F&#x3E;. Here, Td

is a d-dimensional torus,

is a transitive

of

the

group on d

### symbols,

and F is a finite group that acts on each of the coordinate

without

### interchanging

them. F has the same re-

### striction, F,

to any coordinate

The

has

### degree

2d over

C. To estimate the diameters of space forms

where r

we

use the

theorem.

THEOREM 5.1.

### Sn/ Td P(d), F&#x3E;

has diameter at least

cos

where

B = diameter

x

and

### F

is the restriction

### of

F to any coordinate

### S"/(T2d, P(2d), F&#x3E;

has diameter at least arccos

cos B .

### Proof.

We use the law of cosines for

We wish to

### compute

the distance between a well-chosen

in the

### diagonal

and a "farthest" orbit in a coordinate three

=

... ,

+

=

Refer to

4.

(16)

Fig. 4.

### First,

we determine the diameter of the orbit space of a coordinate

### S3

under the action. This will be the diameter of

We

### generally computed

these

diameters in Section 2. We take two

and

on a coordinate

### three-sphere

that realize this diameter. Then the closest

in the

to

is

### clearly

The distance between

and this

is

seen to be

## arccos =

Call this distance C. No other

### point

on any coordinate

### three-sphere

is any closer to this

No other

in the orbit of

### (q, 0, ... , 0)

is

any closer to this

### diagonal point,

either. If we take

where

E

is another

### point

in the coordinate 3-

if

### where q

has been moved to another coordinate three

### sphere.

Neither of these

transformations decreases distance.

### Together, they generate

all the trans-

(17)

formations of the group. So the distance between the orbits of

cosines for

we know that

### A,

the distance from

satisfies

cos A = cos B cos C + sin B sin C cos 0.

The

### angle

0 is that between the minimal

### geodesic

from A to the coordinate

and the

space to the

=

### 03C0/2,

and we can write

cos A = cos B cos C.

As we have

where B is the diameter of the coordinate

### S3

after identifications under

### F.

No element in the orbit of

is any closer to

since for

and

and

### q’

realize the distance between the

orbits. Since

fixes the

### diagonal,

it has no effect on the distance from A to

### Therefore, Sn Td, P(d), F&#x3E;

has diameter at least

cos B.

Where the

has

### degree 4d,

the limit group is contained in

### T2d, P(2d), F&#x3E;,

and the distance C

so A =

cos B.

Wolf’s groups

### give

rise to four basic actions on the coordinate

the

group, and the

### binary tetrahedral, octahedral,

and icosahedral groups. We therefore have four estimates for diameters.

(18)

THEOREM 5.2. The

### quaternionic, tetrahedral, octahedral,

and icosahedral actions on the coordinate

rise to

lower bounds on

the diameters

and

### Proof.

First we consider those

that

rise to the

### quater-

nionic action on the coordinate

Pk,l and ’1k,l.

of

the first

,uk,l are of

2d.

action

### interchanges d

coordinate three-

’1k,1 are of

### degree

4d. If r = Im 03BCk,l, the diameter 2

of

### S"IF /

is at least arccos

### 2 . n+1

The finite group g p

### acting

g on each coordinate

is

P

So each coor-

dinate

is

### subject

to the circle action

### ei03B8I2

and to the action

The diameter

is

The action of

on this

is reduced

### by half, since - I ~ S1. Therefore,

the effective action on the orbit space

### S2

is a

rotation group of order

### four, i.e.,

the dihedral group of

2. It

the

and

### negative

rays of each of the three coordinate axes. The orbit space is therefore one octant of the

### two-sphere

of radius one-half. Its diameter is

Theorem

### 5.1,

we find that diameter

B

is at least

=

Hère 4d = n

so arccos

is at least

= arccos

Here 4d

= n +

so

Lemma

diameter

is also bounded

this number.

if

r = Im ~k,l

diameter of

is bounded

arccos

The

,l

(19)

is 4d.

### Applying

Theorem 5.1 and Lemma

### 1.3,

we find that the diameter of

is

at least

2

### Hère, 8d = n + 1,

so the diameter is bounded

arccos

## n+ 1 2

Now we consider

### representations

whose actions on the coordinate

are

### generated by subgroups

of the circle and the

### binary

tetrahedral group. These are

vk,l, vk,l,j, 03B3k,l,

### 03B6k,l,j,

and 03B3k,l,j. The first two, vk,l and vk,l,j, have

2d. The

others have

4d.

Im

contains

as a

is

induced from

### generator

does not appear as

of the action on

but as

of the

### permutation subgroup P(2d).

The space forms of the

where r is the

### image

under vk,l, vk,l,j, Yk,l’

### 03B3k,l,j, or 03B6k,l,j

of a finite group, have diameter at least arccos

We saw in

Section

diameter of

is

The

of vk,l or

is

2d.

Theorem

diameter of

### Sn/0393

is at least arccos

Here 4d

### 8

so the diameter is at least arccos

### 3(n + 1)

.

Now let r = Im Yk,l,

The

of each

is 4d.

the diameter of

is bounded below

### by

since 8d = n + 1.

### Groups

r whose restrictions to the coordinate

### S3’s

involve octahedral actions have

Since

has

### degree

2d and the diameter of

&#x3E; is

### 1 arccos 1

Theorem 5.1 tells us that the diameter of

is at

least

(20)

of

V and VI are the

### only

nonsolvable fundamental groups of space forms.

contain I* as a

space form

where

r =

or

### Im ki,l,

has diameter at least

The restriction of r to a coordinate

is contained in

has

Table II

(21)

## diameter 2 arccos 73 (tan 10) B y

Theorem

the

of lk,l,j is

### 2d

we have the diameter of

bounded

### by

If r = Im xk,l, its

### degree

is 4d. Therefore Theorem 5.1 and the result of Section 2

### imply

that the diameter of

### S"/Ir

is also bounded below

A

### comparison

with the lower bounds for diameters of space forms in all dimensions

### yields

the theorem of the

### introduction: 1/2 arccos fi ( tan )

serves

as a lower bound on diameters of space forms of all dimensions.

To

we may

### compile

our results in Table II.

References

1. Cheeger, J. and Ebin, D. G.: Comparison Theorems in Riemannian Geometry, American Publishing Co., Inc., New York, 1975.

2. Gromoll, D. and Grove, K.: A generalization of Berger’s rigidity theorem for positively curved manifolds, Ann. Scientifiques de l’Ecole Normale Supérieure, 20 (1987), 227-239.

3. Gromov, M.: Almost flat manifolds, J. Differential Geometry 13 (1978), 231-241.

4. McGowan, J.: The diameter function on the space of space forms, dissertation, University of Maryland, 1991.

5. Michael, E.: Topologies on spaces of subsets, Transactions of the American Mathematical Society

71 (1951), 152-182.

6. Mostert, P. S.: On a compact Lie group acting on a manifold, Annals of Mathematics 65 (1957),

447-455 and Errata 66, 589.

7. Myers, S.: Riemannian manifolds in the large, Duke Math. J. 1 (1935), 39-49.

8. Raghunathan, M. S.: Discrete Subgroups of Lie Groups, Springer-Verlag, New York, 1972.

9. Varadarajan, V. S.: Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, New York, 1984.

10. Vincent, G.: Les groupes Lineaires finis sans points fixes, Commentarii Mathematici Halvetici 20

(1947), 117-171.

11. Wolf, J. A.: Spaces of Constant Curvature, Publish or Perish, Wilmington, 1984.

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