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o

1 (1993), p. 79-98

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

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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

[7],

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

radius

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

[5].

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

[11].

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

restrictions,

and has one additional

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

in case 2 are

already present

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 =

limm~ ~,t~~ 0393mt,

G has an additional

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

radius

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

shaded in

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

to the shaded

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 [10].

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’) [11] .

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

[4].)

We do not answer the

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

general

information about the groups are

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

representations

ak,l have an additional

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

already determined,

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

T*v,

and the additional

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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