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2, Points of Approximation

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§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 half-ray L ending at

L) = 0(1);

(3) for each geodesic half-ray 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 half-rays 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

2lz-vl+(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

half-ray 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 end-point 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 end-point of a

free side is the end-point 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 end-points 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 end-points of a and so is hyperbolic. The geodesic L with the same end-points as a is the axis of h and we

may assume that h generates the stabilizer of L. Note that there is a compact

sub-arc 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 half-ray 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 non-trivial 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 non-zero. 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 G1-orbit. 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 non-elementary 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 end-points

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 half-ray 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 non-conjugate 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

.

(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 non-conjugate

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 non-elementary 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 non-negative 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 non-elementary 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 non-elementary finitely generated Fuchsian group

with signature (10.4.1) and Nielsen region N. Then

h-area(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

h-area(D n N) = h-area(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 end-points

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 +

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