S ÉMINAIRE N. B OURBAKI
M ICHAEL G ROMOV Hyperbolic manifolds according to Thurston and Jørgensen Séminaire N. Bourbaki, 1979-1980, exp. no 546, p. 40-53.
© Association des collaborateurs de Nicolas Bourbaki, 1979-1980, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Séminaire
BOURBAKI
32e année,
1979/80,
Novembre 1979
nO 546
HYPERBOLIC MANIFOLDS
JØRGENSEN
ACCORDING TO THURSTON AND
by Michael
o.
Preliminaries
Surfaces of constant Consider
a
curvature
negative
complete
area)
Vol(V) .
According
We are
important single
interested now in the
It follows that the
x
values of
possible
given volume.
possibilities. V’s
with
V
is its volume (or
V
when this volume is finite. °o
Vol(V)
implies
that
denotes the Euler characteristic.
It is also well known that there a
invariant of
case
to the Gauss-Bonnet theorem the condition
Vol(V) = -2’rnC(V) , where
with
2-dimensional Riemannian manifold
connected orientable
with curvature - 1 . The most
V’s
GROMOV
More
When
are
precisely, k
is
Vol (V )
2 ~r ,
are
4 ’~ ,
...
only finitely many topologically different when
even
Vol ( V ) - 2nk .
Examples
40
Vol(V)
there
are
=
k2
203C0k +
2
and
k
is
odd
there
topological types of
Notice that each surface (with the
points) supports
a
continuum of
exhibits
a
typical
picture
only exeption of the sphere minus three
complete metrics
deformation of
a
-1 . The following
with curvature
hyperbolic
metric
on
the closed surface
of genus two
For the manifolds
the
same
and
Jørgensen,
essential
with
V
properties
as
dim(V) ~
4
for
=
dim
the volume function
2 , but,
the manifolds of dimension three
as
it
was
display quite
V
has
’2014~ Vol(V)
discovered
different
by
Thurston
amazing
features. Manifolds of dimensions We
use
the words
5 ,
4 ,
...
hyperbolic manifold
for
Riemannian manifold with
complete
a
constant sectional curvature -
Finiteness Theorem.- If
Wang’s many
isometry
(see
[W]).
Remarks.-
Wang’s
result in
The number of like
V’s
[W]
with
exp(exp(exp(n + x))) )
x
theorem
Wang’s discrete set
n ~ 4
then for each real
n-dimensional
classes of
implies
is
applicable
Vol(V) ~ but
that for
the real line. When
a
universal constant.
a
Vol(V) = where
is
C
no
fixed
n
to almost all
be
can
x
one has
on
x
hyperbolic manifolds
is n
only finitely
are
with
Vol(V) ,
x .
locally symmetric spaces.
effectively
estimated
(by something
realistic upper bound.
n ~
even
C
there V
3
the values of
form
Vol(V)
the Gauss-Bonnet theorem says
a
more
X(V) ,
n
1. Three dimensional manifolds Thurston’s theorem.- The values of the function all
3-dimensional
hyperbolic
manifolds with
V ~ Vol(V) , where
Vol(V)
°° , form
a
V
runs
over
closed non-discrete uu
set
on
the real line. This set is well ordered and its ordinal type is is finite to one,
function
a given
i.e. there
are
only finitely
volume
Here is
a
schematic
picture
of the values
x =
Vol(V) .
.
many
The
V’s
with
Thurston’s theorem says that there is Then there is the next smallest volume
x1
x3
x2
has
...
limit
a
volume of
and
x , x . We
point
represents the smallest volume of
shall
is
see
The
.
that the number
point
corresponds
x 2
The second statement of Thurston’s theorem says that for each
j =
2 ,
,
Vol(V) =
...
,
f
N
=
=
x
V’s
of different
N(x)
(It is clear that the function
is finite.
x
the number
...
x
manifold. The next smallest
the first manifold with two cusps (see the definition in section 2) and
1
x~ .
forth. The sequence
so
complete non-compact
a
complete non-compact manifold
a
manifold with the smallest volume
a
to
forth.
so
x. ,
with
is unbounded, because
N(x.) J
each
has many non-isometric finite
V
We shall
present
be found in
can
Let
only
some
coverings
5 and 6 of his lectures
chapters
of
a
fixed
degree).
the basic ideas of Thurston’s
mention two unresolved
us
Is the function
Can
below
The details
proof.
[T].
problems.
locally bounded ?
N(x,)
of the numbers
be irrational.
x,/x, 1
J
2. The
Margulis-J~rgensen decomposition
Fix c
be
V
Let
shortest
geodesic loop positive
a
2 1 .~(v) ,
the
e-ball in the
> n
Hn ,
v
E V . When
[K-M]) implies E-ball in
v ~ V
point
at a
even
(Recall, that
-1 . The universal
Hn
covering
is the
complete
of every
with
V
the existence of
with ~ ~ ~
.~(v} ~ 2e .
has
universal
an more
less
or
We shall discuss here
case.
There exists
(special case).-
hyperbolic manifold
generate in
V a
and each
free Abelian
a v
positive number f V
the
subgroup
such
loops based
at
that v
of rank at most two.
[K-M], [T]).
(See
This theorem shows that each a
of the
length
Hn ).
theorem (see
theorem
for each orientable
in
£-ball around
dim(V) .
n =
0 ,I such that each
3-dimensional
length ~ 2~
v C V , the
v .
manifold with curvature
standard geometry,
of
,~ (v ) ,
by
geometry of this ball is standard ; this ball is isometric to the
Kazhdan-Margulis
Kazhdan-Margulis
Denote
and look at the
number e
manifold is isometric to
constant w - w
only the
based at
hyperbolic space
simply connected hyperbolic The
hyperbolic manifold.
a
hyperbolic space
H3
or
These manifolds
in can
a
be
£-ball
£-ball £
~
hyperbolic
manifold with fundamental group
explicitely
in
V
is isometric to
described
as
an
S
follows.
Cusps Take T
x
~~
a
flat two torus
with the metric
T
e-rds2
and +
let
dr2
ds~
denote its (flat) metric. The
is called the double infinite cusp
product C =
C
based
T .
on
It is easy to see that
C =
The manifold has the
boundary
T
(the isometry type of) Here is the
and the
T x 0
=
n
manifold with
and that
is isometric to
7L + ~
=
C C
[0,°o)
T x
hyperbolic
a
3-manifold with
hyperbolic
every
is
C
isometry type
C
of
double infinite cusp.
a
is called the cusp based
This manifold
T .
on
uniquely determined by
is
T .
picture of
cusp for
a
2
n =
Notice, that the double infinite cusps have infinite volume but the cusps have finite volumes. Tubes Consider
3-dimensional
a
that there
see
a)
are
only
Our manifold has
cyclic covering
two
of
- ~ .
n
It is not hard to
possibilities.
closed
no
manifold with
hyperbolic geodesics.
In this case it is isometric to
an
infinite
double infinite cusp.
a
b) The manifold has
closed
a
geodesic.
In this case we call this manifold an
infinite tube.
~ ,
An infinite tube
which is called the axial Let us introduce v
~ 2~
with
The
is the
T =
contractible in
This
the
is
equal
that
geodesic
r
~
03C01
length
(T)
when this
It is not hard to see that the
(T,T) . T C T
~9
.
is
a
simple
a
tube
D
This tube is contracted
r-neighbourhood
o~r
the set of the
as
around
r
length
is
points
as
follows. Take .
The number
E T
T
which is
up to rotations of
length
hyperbolic plane.
of
T , T
It follows
large.
Dr
of
isometry type
r
T
geodesic
Notice that the
circle in the
with ~Dr ~
Y .
closed
uniquely determined,
one can also show that for each flat torus
there exists
Y
geodesic
torus and the induced metric in
topological
~ 03C01(Dr) = Z .
of the radius
log(length(T))
r ~ 0 ,
containes
T
homomorphism to the
closed
r .
torus
by
unique simple
geodesic.
is, clearly, flat. The .
a
(finite) tube
a
dist(v,y)
boundary
clearly, has
T
and
is
such that
T
uniquely simple
a
T
determined
closed
geodesic
is contractible in .
a r
geodesic Y
£
H3
and the
is chosen such that the
by
length of
the
radius
hyperbolic
flat
S
cylinder
isometric and
-
r
We call such
When
(
a
When
V
~
a
is
=
on
(T,T).
a
tube based a
tube
~(v)
the value
T x ]R
cusp
|2r|t
depends only
[0,°°) .
and when is
V
a
or
r(v)
on
When
r
extends to
uniquely
loop
r
~~
we
cusp, the function £
=
we
a
r 2014~ -°°
When
structure. 1 and
simple
very
depends only
l(v)
=
~
r
on
~(r)
the function
have
and £
dist(v,y)
0
=
v ) has
at
~(t,r)
=
infinite tube with the axial
an
is the
=
r
Dr
a
the shortest non-contractible
a
This action
T .
strictly increasing function of -r .
When
~
T/S
T
It follows that there is an
length(03C4) .
=
as
1
is
is about
r
length(S ) such that
T
length(T) . The boundary
equals
manifold is isometric to
length of
=
on
o&r
get
we
with
JR
x
%-action
circle
r
have
is
and for
length(Y)
=
~(v)
the function
Y
geodesic
function of
strictly increasing
a
r
we
have
2r .
function £
In the case of a cusp the
~ (x)
are
tori. When
all levels 1
(x) ,
x
X
>
Using these remarks
picture
is
V
Decomposition theorem.-
V
Let
be
an
constant.
theorem
points where £
=
length(Y)
obtain
a
rather
we
1 2
e
Then the
no
closed
geodesic
of
as
length
a
positive
In this case
E .
2014 (jL
e
V
hyperbolic
consists of
V
follows. Fix
complete
denotes the
where
,
(if any) components and each of these components is isometric to This theorem works
and
is small.
orientable three dimensional
0
manifold of finite volume and let
Kazhdan-Margulis
X
critical value
Kazhdan-Margulis
at the
V
critical values and all levels
no
one
tori.
are
and the
of the geometry of
has
tube there is
a
j
finitely many
a
cusp
such that
and
or
to a tube.
V
containes
are
V.- L ~ )
bounded
by
tori. The
£-neighbourhoods (in
hyperbolic balls. balls of radius
It follows that
connected and hence, The manifold it is
a
V.
length ~ " ~ 2 E
can
C
Vcovered
be
by *
isometric to the
are
N
const E Vol(V)
=
that
is connected.
V
Clearly, the
its diameter is bounded from above has
(0,~] -,
a
more
V
.
complicated
object.
does not exceed
=
2
local geometry than
particular,
In
2N
by
is also
set
const
one
can
see
V- [E,oo) . "const"
Vol(V) , where
~
but,
that the number
is of
.
It follows that
when
.
rather standard
of the components of the order n
V,- !E,oo)
v
e .
We assume,.as usual,
globally,
V ) of the points
(they
V
containes at most
serve
is less than the
as
the axial
length
const
geodesics
Vol(V)
of closed
of the tubes in
of the shortest of these
geodesics
V (0, E j -.
of
) and
geodesics, the manifold
V
consists
(O~~j -.
of cusps and their number does not
only
Here is a schematic
is
diffeomorphic
[M])
the fundamental group
isometry. Thus, there
bounded
compact manifold
By the Mostow theorem (see up to
°
by
following tori and
V’s .
determines
Ti (V)
V
only countably many isometry types of
are
V’s .
Convergence of manifolds For two metric spaces
.
Consider
xi
a
topologically different
are
E
theorem is the
many
uniquely, 3.
to the interior of
decomposition
only countably
hence, there
on
picture
One of the immediate corollaries of the
V
depend
a
and
X , Y
a
map
sequence of metric spaces
E X . We say that the sequence
numbers e > o ,
exists
a)
a
f
map
,
a r
number j , ball
B c
Xl
set
2 ,
...
(Y,y)
,
with base
if for
such that for each around
x
into
arbitrary
i ~ j Y
points there
such that
1
f(x.) = L(f) ~
i = 1
converges to
0 , there is
from the radius
X1 ,
we
Y
b) the image c)
r >
f : X ~ Y
f(B)
C
Y
contains the ball of radius
r- E
around
y E Y
e .
Example Look at the
following
sequence of
compact hyperbolic surfaces of genus
two
In this limit process we have lost
the set
was
V,- L~~°°).
one
disconnected,
which takes into accent all components of
3-dimensional
hyperbolic
(J~rgensen).3-dimensional is
a
decomposition
V
sets one
3-manifold with
can
also
see
that if
(Chabauty). manifolds of then
our
a
lim
(Vl,vi) ,
Let
a
be
convergent sequence of
a
Then the limit space
V
i lim Vol(V ) .
that for
sufficiently small £
a
V- ,
have
hand, for the
Using -.)
the --~
> 0
decomposition 0 .
the
theorem
Since
Vol(V) .
fact
fixed dimensions.
sequence has
Remark.- This
we
because
does not show up.
Val(Vl)
.
then
-L
simple general
a
implies
Q ,
~
const, Recall
Vol(V) =
theorem
E.
.
happend
refined notion of convergence,
On the other
VC~~~) . sup i
surface. This
our
more
complication
connected and
are
a
i - 1 , 2 , ... ,
manifolds with
hyperbolic
hyperbolic
Proof.- The
manifolds this
(Vl,vi) ,
Let
half of
and there is
i=1,2,...,
be
a
sequence of
If the geodesic loops at
convergent subsequence.
compactness criterion holds for
a
(See
v.
satisfy
hyperbolic inf
L(vi)
~W~).
sequence of
arbitrary complete
>
0 ,
Riemannian manifolds with sectional curvatures
pinched
Corollary (J~rgensen).- The values of
form
Proof.-
3-dimensional
all
over
runs
to the
According
,~(v) > W .
Vol(V)
hyperbolic
theorem each
decomposition
V’s
and
Opening
of
hyperbolic
Let
be
V
v f V
point
a
with
convergent subsequence and
a
is continuous.
convergent sequence (relative
a
when
~t~
Q.E.D.
the cusps
closing
V
has
V
has
by the Jqsrgensen theorem the function
4.
closed set in
a
manifolds of finite volume.
It follows that each sequence of
.
between two constants.
3-manifolds with
VOl(Vi)
sup i
to
~ ,
some
choices of the base
and let
points)
denote the limit
V
manifold. It follows from the
i
sufficiently large
vC~ . ~)When .
E
components
as
all
(and °
topology,
into cusps in
we
i
is
are
pass to the
cusps
"
3-manifolds with
cusps and
of
q
a
theorem
a
(0,~j-. (by making E smaller
if
is the
V
turning
to
number of
same
necessary)
follows, that the only possible
~
has
V q
that
change , in
of the tubes of
a
sequence of
hyperbolic orientable a
O
such that each
geodesics of length ~
Ei
with
cusps and it is
p + q
V1 ,
subsequence
E. ~
sequence
p
and
diffeomorphic
V1 q
which
has
p
independent
to each of the
geodesics.
picture above
It is worth
Take
bounded volumes. Then there is
positive
closed
simple
minus these In the
diffeomorphic
have the
V
and for each
following
(J~rgensen).-
uniformely V, and
i , The manifold
’s
> o
V -..
cusp opening
converges to
It
E
are
the sets
large)
limit v, 1
The formal statement is the The
the sets
j(E) )
We can also assume
components in when
i ~
(i.e.
is small
theorem that for each
decomposition
we
mentioning
had that
p a
=
0 ,
q
=
1
.
tube minus the axial geodesic is
diffeomorphic
to
a
T
and
cusp. Moreover, C T
T.
is
one
easily
can
sequence of
a
~
based
geodesics
(T,03C4i)
on
is the cusp based
C
closed
simple
then the sequence of tubes
with
flat torus
a
on
length(T. ) i ~
converges to
--~
~n
(The base points
C.
taken at the boundaries of these tubes).
are
It is not
happend
selves
The second
have
a
boundary)
Vol (Vi)
Vol(V)
and
whithin
occur
Vl
Vol(Vi)
V
when
-~
V .
not isometric to
are
In such a case
be, a priori, equal.
them-
are
V.
(i.e.
can
can
zero, and hence all
always
Mostow’s theorem isometric) to
non-trivial limit of
a
C
" . In other words it could
V )
would
we
discrete set of the volumes.
Both
problem
Closing the
which has
resolved
are
V
cusps.- Let
p + q
--[201420142014201420142014 V~ has
exactly
by
the
be
a
represented by
B.
The volume limit theorem.- Let
q >
theorem and let This theorem
Then
0 .
implies
"
a
limit.
(p+q)-fold"
Vi
~
be
V
a
Vol(V) >
201420142014201420142014201420142014201420142014
particular,
In
V
can
2 ,
,
...
is of the
,
.
is well ordered, because must stabilize. This
...
is finite to one, and
Vol(V)
cusp opening
in the
Vol(V)
Vol(V2) z
V’2014~ Vol(V)
that the set of the values
as
sequence i = 1
with
also shows that the function see
2014201420142014201420142014
geodesics.
that the set of the values
V
each convergent sequence
m
compact manifolds.
limit of
be
a
is
V
Vol(V)
above, such that each
as
sequence
short
q
that
implies
a
..-
cusps and
p
theorems of Thurston.
following remarkable
complete orientable manifold’with
Then there is
cusps.
’
This theorem
we
is
q
Vol
is the behaviour of the volumes
problem
is
V
the volumes
manifolds with
that in theorem above
diffeomorphic (and by
Even when
Di~
clear, however, why the limit process
(i.e. without
complete have
A.
that if
see
type
theorem A
by using
Thus Thurston’s
.
theorem of section 1 is reduced to A and B.
Theorem B is
special
a
of the
case
B’. Thurston’s rigidity theorem.- Let
manifold of finite volume be
the
holds iff
equality
a
is isometric to
1
.
If
q >
V’
0
to
V’
V’
that
was
is
If
and
are
implies
implies
V’ to
that
the Mostow
complete hyperbolic
*
closed
geodesics. d
of
covering
Let
Vol(V’)
and
d
degree
(i.e.
). In particular the equality
complete (and
diffeomorphic
Vol (V)
complete hyperbolic
d . Then
isometric
an
covering of
theorem B’
show that B’
V’
homotopic
a
disjoint simple
some
B’ = B . We take for
Let us
V
is
implies how
We know that
geodesics. degree
explain
f
manifold with
hyperbolic
a
positive degree
d-sheeted
a
Vol(V) = d Vol(V’) Let us
by deleting
proper map of
f : V --~ V’
V
be
V
denotes the manifold which is obtained from
V’
and let
following.
a
geodesics
no
Vl
manifold
V , and
we
so
were
minus
have
our
deleted). q
short
f
of
Vol(V) > Vol(V’) -
rigidity
theorem.
manifolds with
isomorphic
fundamental
Vol(V’)
Vol (V) V
V’
--~
has proper maps
one
groups
is
(otherweise,
homotopic
5. The cusp
to an
We shall discuss
closing
by the
only
simply
the
simplest
says that
flat torus
We know that
If
we
replace
i
degree
We can assume that
.
), and B’ says that the map
V’
and
V
interplace
we
of
V
---;
isometry.
is
C
the cusp
a
C
V
VE=L
T =
of
case
is
V
By the decomposition theorem bounded
V’
and
theorem
closing
theorem
V -~ V’
be divided into
can (E)
by
a
tube i~
based
compact piece
a
C = V
on
V~T.~ .°° ))
length(T,) i get
we
Vl
V =
based
(O.c~j ~
with
on
The manifolds
V .
manifold.
hyperbolic
(T,T i)
on
cusp, when the cusp
one
and the cusp
limit of tubes based
of compact manifolds such that
with
V
a
limit of compact
a
are
a
V
sequence
hyperbolic :
not
~
Vl
the natural metrics in metrics
singular,
are
Since
-c
In order to eliminate
have curvature
in the
-1
outside of
but at
T
these
T
following picture.
this
singularity
is
this
singularity
and to make
appropriate
must construct
one
as
getting "smaller
and smaller"
and
V
some
o
deformations of the
hyperbolic
i ~ ~ .
as
Dii
of
metrics in
fit at
T
and in
Vo
Di. It is not difficult to visualize all
because with
a
we
have
(much
The
explicit description
an
more
hyperbolic
the
deformations when
volume and
Weil’s
stated
V
4 . The last
is
a
tubes. Thus
we
a
tube
left
are
non-trivial deformation
of
the
by
holonomy represen-
observe, that this representation (and hence,
may have
compact
non-trivial deformations. For
no
and
dim(V) ? 3
rigidity property (this
is
or a
when
special
V
example,
has finite
case
of
rigidity theorem) plays the crucial role in Wang’s finiteness theorem
in §
o.
Let us return to our
V . o
determined
in the group of the isometries
V .
underlying hyperbolic manifold) no
essentially
T -
hyperbolic space covering
there is
A.
is
V
tation of the fundamental group of the
hyperbolic
serious) problem of constructing metric in
deformations of
possible hyperbolic
of all
3-dimensional
case.
Notice first that the group of the
orientation
transformations of the
algebraic group form
H3
isometries of
preserving of the
sphere
S2
complex
dimension
is identical with the group of the conformal
of the
r
admits
with
presentation
a
r
representations
generators and 1
k
is of
-~-~
complex
a
--~
~(k -.~- 1) ,
at least
i.e. the
conjugations
When
is
V
because in
2-cells.
of
k- 1
generators with mations. However,
only the trivial deformations,
positive
’~~(V)
can
only
cusp, the Mostow
one
generated by
---~
corresponding
fixed
a
slightly
is manifold
more
assumed to be
theorem
r
~
that the
implies
group ~ ® ~ -
small deformed
arbitrary
point
can
not be
H3/T’
H~ ,
r’
of the
which has
invariant
an
corresponding represen-
us
to
by replacing
V
represent each
in this form,
generalization
V
to the
i.e.
to
equip
a
tube.
from above ( i
it with
original singular
by
C
a
is
hyperbolic
metric in
to several cusps can be found in
Vl .
chapter
[T].
rigidity theorem that the
Euclidean ball
geodesics from if the
Hn
hyperbolic space
is
projectively isomorphic
to the open
i.e. diffeomorphism Hn --~ Bn which sends Hn onto straight segments. A set E C Hn is called a straight corresponding 1 C B is a usual Euclidean simplex. there is
following elementary
For each
where the last represenof
is obtained from
is, automatically, close and the
isometry image
representations
~ PSL(2,C)
cocompact group without torsion and the
careful argument allows
details,
6. Thurston’s
free
that the
discrete
V’ =
5 of Thurston’s lectures
Recall,
a
sufficiently large)
structure which
The
of the
representation
It is not hard to see
geodesic.
simplex
representation
rigidity
which factor as ZZ~ZZ ~ 2Z
tation is
The
Thurston shows
and discrete.
follows, that there exist
r
F ,
defor-
V ).
ZZ+ZZ ~ PSL(2,C)
tation
gives
no
is not less than the number of
general, this dimension
dimension (in
k
presented by
rearrangement of the relations in
a
restriction of the deformed
It
cells of dimension
k
be
relations and the crude estimate from above
by using
In the case of
injective
has
decomposition F =
It follows that
that the space of the non-trivial deformations of the
the cusps of
and
3-manifold with finite volume its Euler
and the minimal cell
zero
1
k -1
given representation
have to factor out
non-compact hyperbolic
a
3 ( k - .~ )
is of dimension
PSL(2,C) .
characteristic is and
we
a
relations the space
dimension at least
complex
the space of small non-trivial deformations of
A
r
3 , and the representations
complex algebraic variety.
a
When
has
that is
i.e. with the group
k,
fact
2 ~ k ~ n ,
the
a
plays
the crucial role in Thurston’s argument.
hyperbolic
volume of
a
k-dimensional
straight
simplex 0394 ~ Hn i,e. the
is bounded
k = 2
When
corresponding 11
The maximal showed (see is
infinity
7 in
[T])
of this
C3
has all three vertices at
infinity,
Sn-~ -
The
to
simplex
boundary
Notice that all
n .
C2 -
.
2-dimensional
straight
isometric.
are
that this
projectively equivalent
The volume
Hn
simplex A
3-dimensional
chapter
Ck .
has the vertices at the
’n , i . e .
is
with vertices at
constant
a
simplex
Bn
C
volume (i.e. area) of A
simplices
by
the maximal
also has the vertices at
regular, i.e.
is
simplex
the
infinity. Milnor
corresponding
regular Euclidean simplex with vertices
a
is
C3 =3 203A3 1 2sin(203C0i 3) ~
given by
Sn-1 .
at
1.0149
(see
[T]). When
This is
infinity. Let
and
the maximal
k ~ 4
us
recent result of
a
emphasize
that there is
k ~ 3 . All ideal (i.e.
and have the between
C .
are
also
Haagerup
regular simplices with
and Mankholm
with the vertices at
infinity)
intimately
vertices at
[H-M]).
related to the
k = 2
cases
2-simplices
the volumes of the ideal
k ? 3
The last fact is
(see
difference between the
principal
a
volume, but when
same
and
0
simplices
regular
are
simplices
vary
rigidity phenomena
in dimensions ¿ 3 . Notice also that
(see
have the
C~
when
k
[H-M]). A map from an Euclidean
this
following asymptotics
simplex
simplex homeomorphically A map from
to the universal A map from
into
s
Let
sional
be
homotopic The
to
a
straight simplex manifold
Hn )
straight.
=
is
simplex
is
is called
into
V
H
in
is called
V
if it sends
straight
is called
if its
straight
straight
lifting
if the restriction
straight.
fact is obvious. m-dimensional
an
hyperbolic
a
simplicial polyhedron
following K
onto
hyperbolic
covering ( a
of this map to each The
a
Hn
into
s
straight
following
simplicial polyhedron n ~
manifold with
and let
V
Then every continuous map
m .
be
n-dimen-
an
K --~ V
is
map.
result
can
be viewed as
a
crude version of Thurston’s
rigidity
theorem.
Thurston’s mapping theorem.- Let There exists
a
constant
hyperbolic manifold
Proof.- Fix
a
C =
M
C(M)
and for
an
triangulation
of
be
a
arbitrary
M
closed oriented
such that for
and
an
arbitrary
that
f
oriented
f : M -~ V
continuous map
assume
n-dimensional manifold.
is
straight
one
n-dimensional has
relative to this
Let
triangulation.
n-dimensional
denote the
s~,...,s~
simplices
of this
triangu-
lation. We have
Corollary.-
a
2
explain
Let us
compact orientable hyperbolic manifold. Then
f : V ~--~ V
If
Proof.-
f
then the iterates of
how these ideas
straight simplices. and let
triangulation
an
arbitrary
satisfies
be
can
have
Let
applied
s ,...,s.7
(1 - ~i ) , in
i=
V’
and
V
n-dimensional
be the
v. =C
arbitrary large degrees.
to Thurston’s
compact oriented hyperbolic manifolds
Take two
into
be
V
Let
continuous map
1,...,j,
theorem.
rigidity and
triangulate
simplices
of
V
this
denote the volumes of these
simplices. For
a
.
could make
we
Unfortunately, simplices (they ideal
are
we
E - ~
---~
we
have
have
would get
we
usual
there is
no
the
ones
only
V’
Vol(V)
deg(f) s Vol(V’)
triangulation consisting
with volume
Cn
), but
one
of infinite
regular
instead
can use
some
triangulations. Denote
the set of all ideal (i.e. with the vertices at
S
by
n-dimensional
simplices
the set of the of
f : V
map
rnax E.. Then ~i
E =
Let
If
straight
in the universal
covering
One views the set
regular simplices.
Hn
V = R
infinity) R C S
and let
as
an
ideal
denote
triangulation
V . Denote
S’
by
One can show
direct
geometric argument
pullbacks
=
n
When
Vol(s),
and that the
deg(f) , n ~ 3
homotopic
R’
the
as
of the Borel sets C
d =
and
(and hence
to an
sets of
corresponding
isometric
in are
s
[T],
measurable) map
E R ,
equality
f
that the map
one
can
f : R
holds iff
f
See
chapter 6
--~
[F])
of
f
noncompact manifolds with finite volume.
by
a
S’ . Using the inequality
implies
[T]
or
measurable (i.e, the
sends almost all
,
is valid for the
a
Vol(V’) S d
show that
S’ ~ R’ ) this property of
covering.
induces
V’ .
associated to
simplices
by using Furstenberg’s boundary construction (see
Vol(V) , R
into
R’ C S’ .
that the map
for the actual
proof
f
is
which
REFERENCES
[C]
C.
CHABAUTY - Limite d’ensembles et géométrie
des
nombres, Bull.
Soc.
Math.
France, 78(1950), 143-151.
[F]
H.
FURSTENBERG -
Sump. in
[H-M]
U.
Boundary theory
Pure
HAAGERUP, H.
and
stochastic processes
on
homogeneous
spaces,
Math., Vol. XXVI, 1973, 193-232.
MUNKHOLM - Simplices
of maximal volume in
hyperbolic
N-space,
Preprint.
[K-M]
D.
KAZHDAN, G. MARGULIS,- A proof of Selberg’s
hypothesis,
Math. Sb.,
(117)
75(1968), 163-168.
[M]
G.D.
MOSTOW -
Strong rigidity
of
locally symmetric
spaces, Ann. of Math.
Studies, Vol 78, Princeton 1973.
[T]
W.
THURSTON - The geometry Princeton
[W]
H.C. WANG -
and
topology
of
3-manifolds,
Lecture Notes from
University, 1977/78.
Topics in totally
éd. Boothby-Weiss,
N.Y.
discontinuous groups, in
"Symmetric spaces",
1972, 460-485.
Michael GROMOV, ,
State University of New-York at Mathematics department Stony Brook NEW-YORK 11794
53
Stony Brook
