2, Points of Approximation
Tải bản đầy đủ  0trang
§10.2. Points of Approximation
259
this fastest rate can be described in
are asymptotic to each other as n —*
hyperbolic terms, namely,
= O(1/cosh p(O,
—
or in terms of matrices, namely,
K
—
=
Moreover, it is easy to see that we can replace the origin in the first two
expressions by any z in A. This is implicit in the next result which provides
yet another interpretation of this fastest rate of convergence.
Theorem 19.2.1. Let G be a Fuchsian
acting in A, let be a limit point
be distinct elements of G. Then the following stateof G and let gj, g2,
ments are equivalent:
(1) JbreachwinA,
—
K
=
(2) for each w in A and each geodesic halfray L ending at
L) = 0(1);
(3) for each geodesic halfray L ending at
such that for all n,
there is a compact subset K of A
0.
PRooF. In general, p(gw, L) m if and only if g  1(L) meets the compact
disc {z: p(z, w) m}: thus (2) and (3) are equivalent and for a given L, (2)
is true or false independently of the choice of w. Further if L1 and L2 are
geodesic halfrays ending at then for some m1,
L2
{z:p(z,L1)
so (2) is also true or false independently of the choice of L. For the remainder
of the proof, L will denote the Euclidean radius [0, and L' will denote the
then
Observe that if z is close to
Euclidean diameter (—
p(z, L) =
p(z, L').
Suppose first that (1) holds. Then putting w =
K
This implies that
—*
—
= 0(1
0
we obtain
—
so for all sufficiently large n,
L) =
L').
260
10. Finitely Generated Groups
If z e A then (from Section 7.20) we have
sinhp(z, L) =
—____
— Iczi
—
I —
2jz
I
—

Cl
lzI
Putting z =
with n large, we obtain (2) in the case w = 0. As (2) is
independent of the choice of w, we see that (1) implies (2).
Next, let z be in A and closer to C than to — C and let v be the foot of the
Euclidean
perpendicular from z to L'. Then C
is
the point on ÔA that is
nearest to v and so
lz
 Cl lz  vj + lv  Cl
z  vi + lv  (z/lzl)j
vi + Jz  (z/lzDl
212
2lzvl+(1 lzD.
As
lz — vj
we
= Im[Cz]j,
deduce that
IZCI
2 sinh p(z, L') + 1
2 sinh p(z, L) + 1.
Putting z =
w = 0.
and using (2), we find that (2) implies that (1) holds with
Finally, if w E A and
g(z) =
az+ë
cz
al2 — ci 2 = 1,
—,
+ a
we obtain (by direct computation)
g(w)  g(O)j
(1

wj)coshp(0,gO)
(10.2.1)
(1 because
al =
cosh
gO),
Ad = sinh
We have seen that (2) implies that (1) holds when w =
(10.2.1) yields (1) for a general w.
gO).
0.
Clearly this with
§10.2. Points of Approximation
261
In view of the different characterizations of the fastest rate of convergence
it is convenient to adopt some suitable terminology.
Definition 10.2.2. A limit point of a Fuchsian group G is a point ofapproximation of G if for each win A there is a sequence of distinct g, in G with
I
—
=
11—2).
Theorem 10.2.3. A point of approximation of a Fuchsian group G cannot lie
on the boundary of any convex fundamental polygon for G.
PROOF. Suppose that a point of approximation lies on the boundary of a
convex fundamental polygon P. By convexity, we can construct a geodesic
By Theorem 10.2.1(3), the images
halfray L lying in P and ending at
'(P) meet a compact set and this violates the fact that P is a locally
finite (see Definition 9.3.1).
Example 10.2.4. Every parabolic fixed point of a Fuchsian group G lies on
the boundary of some Dirichiet region: thus a parabolic fixed point of G
cannot be a point of approximation of G.
For a finitely generated groups, Theorem 10.2.3 and Example 10.2.4
give a complete description of the limit points of G.
Theorem 10.2.5. A Fuchsian group G is finitely generated and only jf each
limit point is either a parabolic fixed point of G or a point of approximation
of G.
Remark. Let us say that the limit set A splits if it contains only parabolic
fixed points or points of approximation of G. If G is finitely generated, then
there exists a finite sided convex fundamental polygon for G (because (1)
implies (3) in Theorem 10.1.2). We shall show that the existence of such a
polygon implies that A splits. We will also prove that if A splits then every
convex fundamental polygon for G has finitely many sides and this implies
that G is finitely generated (because (4) implies (1) in Theorem 10.1.2).
Observe that this reasoning shows that Theorem 10.1.2(3) implies that A
splits and hence that Theorem 10.1.2(4) holds. Thus in proving Theorem
10.2.5 in this way, we also complete the proof of Theorem 10.1.2.
PROOF OF THEOREM 10.2.5. First, suppose that A splits and let P be any
convex fundamental polygon for G. If P has infinitely many sides, then these
sides must accumulate at some point C on
As the Euclidean diameters
of the images of P tend to zero, must be a limit point on 3P. By Theorem
10.2.3, cannot be a point of approximation of G and by Theorem 9.3.8,
C cannot be a parabolic fixed point of G (else.two sides of P end at C), This
contradicts the fact that A splits so P can only have finitely many sides.
262
10. Finitely Generated Groups
Now suppose that P is a finite sided convex fundamental polygon for G:
we may assume that P is a Dirichiet polygon (as the proof of Theorem
10.1.2 shows that in this case, G is finitely generated and then any Dirichiet
polygon is finite sided) and we may assume (for simplicity) that the conditions stated in Theorem 9.4.5 hold. By conjugation, we may also suppose
that the centre of P is at the origin.
If two sides of P, say s and s', have a common endpoint v on 8A, then v
is a parabolic fixed point of G (Theorem 9.4.5) and the stabilizer of v is
generated by a parabolic element p of G which maps s onto s'. Now construct an open horocyclic region at v bounded by a horocycle Q. Note that
there is a compact arc q of Q such that Q is the union of the images pfl(q),
n a 1.
A similar construction holds for the free sides of P. Each endpoint of a
free side is the endpoint of some image of some free side. The interval of
discontinuity a in which a given free side lies is the countable union of
images of the finite number of free sides of P: these images are nonoverlapping and accumulate only at the endpoints of a. It follows that some
h in G maps one image of a free side in a to another such image, also in a,
and so h(a) = a (because the intervals of discontinuity are permuted by the
elements of G). We deduce that h fixes both endpoints of a and so is hyperbolic. The geodesic L with the same endpoints as a is the axis of h and we
may assume that h generates the stabilizer of L. Note that there is a compact
subarc I of L such that L is the union of the images
nE1
The geodesics L and the horocycles Q are finite in number and they
from a compact subset P0 of P.
separate the boundary points of P on
Let K denote the compact set consisting of the union of P0 and the finite
number of arcs q and I.
Now let be any limit point of G which is not a parabolic fixed point and
let L0 be a geodesic halfray ending at The initial point of L0 can be mapped
to a point in P and the corresponding image of L0 cannot lie entirely in one
of the horocyclic or hypercyclic regions constructed above else it ends at a
parabolic fixed point or an ordinary point of G respectively. It follows that
either L0 meets P0 or, alternatively, L0 meets one of these regions in which
case some image of L0 meets one of the arcs q or I. In both cases an image
of L0 meets K and so there is some in L0 with, say, g0(z0) in K.
be obtained from L0 by deleting the initial segment of L0 of
Now let
in K.
contains some Zn with
length n. Exactly as for L0, the ray
Clearly,
+ and the set {g1, g2, . .} is infinite: thus by Theorem 10.2.1,
is a point of approximation and A splits.
.
EXERCISE 10.2
1. Verify Example 10.2.4 by working in H2 with
Theorem 10.2.1(2)).
the parabolic fixed point (use
§10.3. Conjugacy Classes
263
§10.3. Conjugacy Classes
Any group is partitioned into the disjoint union of its conjugacy classes.
The classification of conformal Möbius transformations is invariant under
conjugation and so we may speak unambiguously of elliptic, parabolic and
hyperbolic conjugacy classes. Within the group of all Möbius transformations, the conjugacy classes are parametrized by the common value of trace2
of their elements but, as we shall now see, this is not true of the conformal
group of isometries of the hyperbolic plane.
Theorem 10.3.1. Within the group of all isometries of the hyperbolic plane,
two nontrivial conformal isometries are conjugate and only they have the
same value of trace2. Within the group of conformal isometries, the value
trace2 determines two parabolic or elliptic conjugacy classes or one hyperbolic
conjugacy class.
PRooF. We shall prove the result in detail for the parabolic case only. Using
the model H2, any two parabolic isometries are conjugate (in the group of
conformal isometries) to, say, z z + p and z
z + q where p and q are
real and nonzero. These are conjugate in the group of conformal isometries
if and only if for some real a, b, c and d with ad — bc = 1, we have
a(z+ p) + b
az + b
c(z+p)+dcz+d
+
q.
we find that cp = 0: thus c = 0 and op = dq. As ad = 1,
q so p and q must have the same sign. This shows that within
the conformal group, trace2 determines two conjugacy classes of parabolic
elements. In the full group of isometries, however, the translations z z + 1
and z z — 1 are conjugate: indeed if a and /3 denote reflections in x 0
and x = respectively, then
fia =
are conjugate.
and
so
Putting z =
—d/c,
we have a2p
The elliptic case is handled similarly using the model A and two
rotations fixing the origin. The hyperbolic case is best handled in H2 with
In this case, each element is
two hyperbolic elements fixing 0 and
conjugate to its inverse because there is a conformal isometry, namely
z i—+
— 1/z,
We
E
interchanging 0 and co.
are now going to examine in detail the conjugacy classes in a Fuchsian
group.
Theorem 10.3.2. Let G be a Fuchsian group and let v1, v2,. . be the parabolic
and elliptic fixed points on the boundary of some convex fundamental polygon
for G. Suppose that generates the stabilizer of vj: then any elliptic or parabolic element of G is conjugate to some power of some g3.
.
264
10.
Finitely Generated Groups
PROOF. If g is elliptic or parabolic with fixed point v, then some h in G maps
v to some point on tiP. Thus for some j,
hgh
we
have h(v) =
and then
D
e
.
Corollary 10.3.3. If G is finitely generated, then G has a finite number of
such that any elliptic or parabolic
maximal cyclic subgroups (g1 >
element in G is conjugate to exactly one element in exactly one of these subgroups.
We need only observe that if g is elliptic or parabolic and if two powers
of g are conjugate, say if
=gm,
then h has the same fixed points as g and so is itself a power of g: thus n = m.
Note that if g is parabolic and fixes v, then h also fixes v and so cannot be
hyperbolic.
Later, we shall need information on the number of such conjugacy classes
of these maximal cyclic subgroups in a subgroup G5 of G and the following
simple result is sufficient for our needs.
Theorem 103.4. Let G be a Fuchsian group and G1 a subgroup of index k in
G. Suppose that G and G1 have t and t1 respectively, conjugacy classes of
maximal parabolic cyclic subgroups. Then t1 kt. The same result holds Jbr
elliptic elements.
PROOF. Let D be a Dirichlet polygon for G in which parabolic and elliptic
have cycle length one. Thus exactly t parabolic fixed
fixed points on
points lie in ÔD. Now express G as a coset decomposition, say
D*=
contains at least one point from each G1orbit. As D* has at most kt para
bolic fixed points on its boundary, we have t1
kt. The same proof holds
for elliptic elements.
We turn now to the conjugacy classes of hyperbolic elements in a Fuchsian
group.
Theorem 103.5. Any nonelementary Fuchsian group contains infinitely many
conjugacy classes of maximal hyperbolic cyclic subgroups.
PROOF.
Suppose not, then there are hyperbolic elements h1,..., h7 in G
such that each hyperbolic element in G is conjugate to some power of some
Let u and v be distinct limit points of G. By Theorem 5.3.8, there are
§10.3. Conjugacy Classes
265
hyperbolic elements
with distinct axes A1, A2,... such that
has endpoints
u and
v.
As each f,, is conjugate to some power of one of a finite number of the
we may relabel and assume that = h1 for every n. Then
.
.
.
=
say,
and so the elements
distinct axes
and the same translation length T as h1. As
converges to the geodesic (u, v), this violates discreteness: explicitly, if z e (u, v),
then
have
sinh
p(z,
+
as ii —*
+
yet the
are distinct.
Now let the conjugacy classes of hyperbolic elements in a Fuchsian
group G be C1, C2
The elements in
have a common translation
length, say
+
Theorem 103.6. If G isfinitely generated then
asn
—+
+
cr0.
PROOF. Theorem 10.2.5 and its proof shows that every hyperbolic fixed
point of G is a point of approximation and moreover, that there exists a
compact subset K of A such that every hyperbolic axis has an image which
meets K. This means that every hyperbolic conjugacy class
contains an
element with its axis
meeting K. For some d,
K
{z
A: p(O,
z)
d}.
From Sections 7.4 and 7.35 we obtain
= 2cosh p(O,
= 2 + 4sinh2
p(O,
2 + 4cosh2(d)
sol,—+ +ccasn÷+oro.
Remark. Using known information about the convergence of series, for
example, Theorem 5.3.13, we can obtain more precise information about
the rate at which tends to + cc.
There are two types of hyperbolic elements in a Fuchsian group which
warrant special attention. First, there are the simple hyperbolic elements
(Definition 8.1.5). There are also the boundary hyperbolic elements h which
266
10. Finitely Generated Groups
are characterized by the fact that they leave some interval of discontinuity
on the circle at infinity invariant: of course, these only exist for Fuchsian
groups of the second kind.
Theorem 10.3.7. A finitely generated Fuchsian group has only afinite number
oJ conjugacy classes of maximal boundary hyperbolic cyclic subgroups. A
finitely generated Fuchsian group can have infinitely many conjugacy classes
of primitive simple hyperbolic elements.
PROOF. A finitely generated group G has a convex fundamental polygon P
with only a finite number of free sides, say s1
Each free side lies
in an interval of discontinuity whose stabilizer is generated by a boundary
hyperbolic element, say h1.
If h is any boundary hyperbolic element, it leaves some interval of dis
continuity a invariant and we can construct a halfray L ending at some
interior point of a and lying entirely in some image f(P) (because the images
1(L) lies in P and
of P do not accumulate at the interior points of a). As
ends at an ordinary point of G, it must end in some
Thus f(a) = and
so fhf' leaves invariant: this proves that h is conjugate to some power
of
Finally, we must exhibit an example of a finitely generated Fuchsian
group which contains infinitely many nonconjugate primitive simple
hyperbolic elements.
Construct a quadilateral P in with vertices v1, v2, v3, v4 lying on the
circle at infinity. Let f and g be hyperbolic elements pairing the sides of P
as illustrated in Figure 10.3.1. By Poincaré's Theorem (see Exercise 9.8.2),
the group G generated by f and g is discrete and P is a fundamental polygon
for G. As J and g pair sides of a convex fundamental polygon, they are
Figure 10.3.1
§10.3. Conjugacy Classes
267
vs
Figure 10.3.2
simple hyperbolic elements of G (Theorem 9.7.1). It is clear from the
geometry of the actions that the axes of f and g cross P and this implies
that f and g are primitive.
Now let v5 = f(v1): then the quadilateral with vertices v1, v3, v4, v5 is
also a convex fundamental polygon for G, this time with its sides paired by
f and fg: see Figure 10.3.2. Exactly as above, f and fg are simple, primitive
hyperbolic elements.
This process can be repeated to obtain a sequence g, fg, f2g, .. of primitive simple hyperbolic elements of G. By conjugation, we may assume that
G now acts on H2 and that
.
ỗ
(u
0
1/u'
ớa b\
d)'
where u > 1. A trivial computation shows that
as
+ +
n —+ + cc so the sequence (fflg) contains infinitely many nonconjugate
elements (note that a is not zero else g and f have a common fixed point). 0
EXERCISE 10.3
1. Construct an infinitely generated Fuchsian group G containing infinitely many
conjugacy classes of simple primitive hyperbolic elements with the same translation
length (see Theorems 10.3.6 and 10.3.7).
2. Verify the details in the text relating to Figures 10.3.1 and 10.3.2 in the proof of
Theorem 10.3.7 (use Exercise 9.8.2). Give an alternative construction in which the
vertices
are replaced by free sides and apply Poincaré's Theorem directly.
268
10.
Finitely Generated Groups
§10.4. The Signature of a Fuchsian Group
Let G be a finitely generated nonelementary Fuchsian group. Any Dirichiet
polygon D for G is finite sided and topologically,
is a compact surface
S of some genus, say g, with a certain number of holes removed. As
and
are homeomorphic (Theorem 9.2.4), the genus g does not depend on
the choice of D.
Now consider the Nielsen region N for G and corresponding quotient
The argument given in Section 10.3 shows that the boundary
space
consists
of all axes of all boundary hyperbolic elements in G. Let
of N in
A be one such axis with stabilizer generated by h and let H be the component
of is—A not containing N. Obviously, H is stable with respect to so the
is topologically a cylinder, namely H/ (Theorem
projection of H into
6.3.3). One end of this cylinder is the simple ioop A/: indeed no image
of A can cross A (as the open arc of ÔA which bounds H contains only
ordinary points of G) and there are no elliptic elements of order two stabilizing A (else G would then have only two limit points).
If we denote the natural projection of onto A/G by it, we see that
the
disjoint union of n(N) together with simple ioops of the form it(A)
is
and with cylinders of the form ir(H). The cylinders ir(H) are joined to rr(N)
across the common boundary loops ir(A) and there are the same number,
say t, of these as there are conjugacy classes of maximal boundary hyperbolic cyclic subgroups. It is clear now that the three spaces A/G, DIG, N/G
are homeomorphic to each other.
In addition, G contains only a finite number, say s, of conjugacy classes
of maximal parabolic cyclic subgroups and each of these corresponds to a
puncture on the surface S (consider the quotient of a horocyclic region that
is stable under a cyclic parabolic subgroup). Finally, G contains only a
finite number, say r, of conjugacy classes of maximal elliptic cyclic subgroups:
let these have orders m1, ...,
respectively. We introduce terminology to
summarize these facts.
Definition 10.4.1. The symbol
(g:mj,...,mr;s;t)
(10.4.1)
is called the signature of G: each parameter is a nonnegative integer and
2.
If there are no elliptic elements in G, we simply write (g: 0; s; t). It is
possible to state precisely which signatures occur.
Theorem 10.4.2. There is a nonelementary finitely generated Fuchsian group
with signature (10.4.1) and
2 and only (f'
2g—2+s+t+
\
mj/
(10.4.2)
§10.4. The Signature of a Fuchsian Group
269
The proof that (10.4.2) is a necessary condition for the existence of a
group with signature (10.4.1) is a consequence of the following result.
Theorem 10.4.3. Let G be a nonelementary finitely generated Fuchsian group
with signature (10.4.1) and Nielsen region N. Then
harea(N/G) =
+s+t+
2
j=1
(i
—
If G is also of the first kind, then N = A arid t = 0: thus we obtain a
formula for the area of any fundamental polygon of G.
Corollary 10.4.4. Let G be a finitely generated Fuchsian group of the first
kind with signature (g: in1
mr; s; 0). Then for any convex fundamental
polygon P of G,
= 22t[2g
—
+s+
2
PROOF OF THEOREM 10.4.3. We take D to be the Dirichiet polygon ior G
with centre w so
harea(D n N) = harea(N/G).
By choosing w appropriately, we may assume that each elliptic and parabolic
cycle on
has length one and (by taking w to avoid a countable set of
geodesics) we may assume that no cycle of vertices of D lies on the axes of
hyperbolic boundary elements.
Clearly, only finitely many distinct images of a hyperbolic axis can meet
the closure of any locally finite fundamental domain. As N is bounded by
hyperbolic axes (because G is finitely generated), this implies that only
finitely many sides of N meet D and so D n N is a finite sided polygon. The
consists of, say, 2n paired sides (which are arcs of paired
boundary of D
sides of P) and k sides which are not paired (and consist of arcs in D of the
axes bounding N). The vertices of .D n N are the r elliptic cycles of length
one, the s parabolic cycles of length one, some accidental cycles of P (say
a of these) and finally k cycles of length two corresponding to the endpoints
of the k unpaired sides of D n N.
Applying Euler's formula (after "filling in" the holes), we obtain
2—2g=(1+t)—(n+k)+(r+a+k+s)
so
n— a
=
2g
—
1
+r+S +
t.