LIMIT POINTS OF KLEINIAN GROUPS AND FINITE SIDED FUNDAMENTAL POLYHEDRA
BY
A L A N F. B E A R D O N and B E R N A R D M A S K I T University of Cambridge State University of New York
England Stony Brook, N.Y. 11790, USA
Let G be a discrete subgroup of SL(2, C)/{• 1}. Then G operates as a discontinuous group of isometrics on hyperbolic 3-space, which we regard as the open unit ball B a in Euclidean 3-space E a. G operates on S 2, the boundary of B a, as a group of conformal homeomorphisms, b u t it need not be discontinuous there. The set of points of S 2 at which G does not act discontinuously is the limit set A(G).
If we fix a point 0 in B a, then the orbit of 0 under G accumulates precisely at A(G).
The approximation is, however, not uniform. We distinguish a class of limit points, called points o/aproximation, which are approximated very well b y translates of 0. The set of points of approximation includes all loxodromic (including hyperbolic) fixed points, and includes no parabolic fixed points. In w 1 we give several equivalent definitions of point of approximation, and derive some properties. We remark t h a t these points were first discussed b y Hedlund [7].
Starting with a suitable point 0 in B a, we can construct the Dirichlet fundamental polyhedron P0 for G. I t was shown by Greenberg [5] t h a t even if G is finitely generated, P0 need not have finitely m a n y sides. Our next main result, given in w 2, is t h a t if P0 is finite-sided, then every point of A(G) is either a point of approximation or a cusped parabolic fixed point (roughly speaking a parabolic fixed point is cusped if it represents the right number of punctures in (S2-A(G))/G).
The above theorem has several applications: one of these is a new proof of the following theorem of Ahlfors [1].
I / P o has finitely many sides, then the Euclidean measure o/ A(G) is either 0 or 4~.
Our next main result, given in w 3, is t h a t the above necessary condition for P0 to have finitely m a n y sides is also sufficient. I n fact, we prove t h a t any convex fundamental polyhedron G has finitely m a n y sides if and only if A(G) consists entirely of points of
1 - 742908 Acta mathematica 132. Imprim6 1r 18 Mars 1974
2 A. F . B E A R D O N A N D B. M A S K I T
approximation and cusped parabolic fixed points. As an application of this we give a new proof of the following theorem of Marden [11].
Every Dirichlet /undamental polyhedron is /inite sided or none are.
w
L e t ~a be the 1-point compactification of E 8, the added point is of course called c~.
Then G acts on ~a as a group of orientation preserving conformal homeomorphisms.
I n ]~a, the unit ball B 3, and the upper half-space
m = {(~, ~)l~eC, ~ e g , ~ > 0 )
are conformally equivalent. When convenient, we will regard G as acting on H a, and on C, its boundary.
I n E a we use
Jx-yJ
for Euclidean distance, and in B a or H a, we use Q(x, y) for non- Euclidean distance.The action of G on ]~a is m o s t easily seen via isometric spheres. We assume t h a t co is not fixed b y
gEG,
and t h a t g(B a) = B a. Then there are two 2-spheres Sg and S~, called theisometric spheres
of g and g-Z, respectively, with the following properties: Sg and S~ both have the same (Euclidean) radius Ro, and are b o t h orthogonal to S ~. The action of g is the composition of inversion in So, followed b y reflection in the perpendicular bisector of the line segment joining the centers of Sg and S'g, followed b y a Euclidean rotation centered at the center of S~. The importance of this description is t h a t g is the composition of inversion in Sg and a Euclidean isometry (which m a p s g-i(oo) to g(oo)).We enumerate the elements of G as (gn}, and let Rn be the radius of the isometric sphere of g~. I t was shown b y Beardon and Nicholls [3], t h a t for every positive e,
~ R~+~<
~ ,while ~ R~ < c~
if G is discontinuous at some point of S 2.
I t is useful to compare R o with Jg(0)l and I g ( ~ ) J (0 is now the origin). As S a and S 2 are orthogonal,
R ~ + 1 = Ig(oo)l ~
and as g(0) and g ( ~ ) are inverse points with respect to S ~, Ig(0)l.
Ig( )l
= 1 . I fG=(gn}
is discrete then ]gn(r and so
89 -0 ]gn(oo)] - 1 N 1 - ] g n ( 0 ) J .
a s n - + o o .
L I M I T P O I N T S O F K L E I N I A N G R O U P S
W e can use t h e above description of g to derive t h e following result, t h e plane ver- sion of which is trivial. I f g is a conformal i s o m e t r y of B a a n d if x a n d y are in E a - {co, 9-1(cr t h e n
R~lx-yl
Ig(x)-g(y)[- ix_ g.l(oo)l ly_ g_, (oo)1 (1)
T h e proof is easy. I f J denotes inversion in S~ we h a v e t h a t ]g(x) -g(Y)[ = [J(x) - J ( Y ) I
a n d also t h a t the triangles with (ordered) vertices g-l(oo),
x, y
a n d g-l(o~),j(y), J(x)
are similar. These facts lead easily to (1).N o w let K be a c o m p a c t subset of ~ ( G ) = ~ a - A ( G ) . I t is easily seen f r o m (1) t h a t there are positive n u m b e r s k I a n d k 2 (depending on G a n d K) such t h a t for all x a n d y in K a n d all b u t a finite n u m b e r of n,
21 R2 ~ Ign (x) -- fin (Y)]'<< k2 R2" (2) A limit point z is called a
point o/approximation
of G if a n d only if there is a point x in ~(G), a positive c o n s t a n t 2 a n d a sequence gn of distinct elements of G withIs-g.(z) I < 2R . (3)
W e r e m a r k t h a t b y (2) this holds for one x in ~(G) if a n d o n l y if it holds for all x in O(G).
F u r t h e r , t h e a p p r o x i m a t i o n (3) is u n i f o r m on c o m p a c t subsets of O(G).
A n o t h e r observation is t h a t t h e rate of a p p r o x i m a t i o n b y points in ~(G) as expressed b y (3) is t h e best possible. I n d e e d if we replace g, x a n d y in (1) b y g~', z a n d 0 we find t h a t
(4)
where 23 is positive a n d depends o n l y on G.
The i d e n t i t y (1) can be used to characterize points of a p p r o x i m a t i o n in a n o t h e r way.
W e p u t y = z in (1) a n d deduce t h a t z is a p o i n t of a p p r o x i m a t i o n if a n d o n l y if for one (or all) x other t h a n z, there is a positive n u m b e r k a n d a sequence g~ of distinct elements of G with
I I 2. (5)
Again, if this holds for some x (~=z) it holds u n i f o r m l y on c o m p a c t subsets of ~,3_ {z}.
I n t h e other direction if (5) holds u n i f o r m l y on a set A we find t h a t z is n o t in t h e closure of A.
The conditions (3) a n d (5) are metrical: we n o w seek to describe points of approxi- m a t i o n topologically. Observe first t h a t if a is a hyperbolic line in B a with e n d points
A. F . B E A R D O N A N D B. M A S K I T
x a n d z, say, t h e n (5) h o l d s for a class of g . if a n d o n l y if t h e r e is a c o m p a c t s u b s e t K of B a w i t h
g.(a) N K # 0 (6)
for t h e s a m e class of g~. W e m a y , of course, t a k e K t o b e {xEBa: ~(x, 0) ~<Q0} a n d w r i t e T = (xEB3: q(x, a) ~<~o}.
W e t h e n see t h a t (6) h o l d s if a n d o n l y if g~(x) -~ z
in T for one (or all) x i n K . A Stolz r e g i o n a t z is a cone in B 3 of t h e f o r m
{xeB': Iz- l k,(l-Ixl)}
a n d n e a r z, T c o n t a i n s a n d is c o n t a i n e d in Stolz r e g i o n s a t z.
W e collect t o g e t h e r t h e a b o v e results.
T H E O R E M 1. The following statements are equivalent.
(i)
(ii)
(7)
z is a point o/ approximation.
For some (or all) x in ~ ( G ) there is a positive number k and a sequence o/distinct elements gn in G such that I z - g , ( x ) l < k . R ~ .
(iii) For some x other than z, there is a positive number k and a sequence o / d i s t i n c t elements gn in G such that ]gn(x)-g,(z)I >~k.
(iv) There exists a sequence g, o/ distinct elements o/ G such that I g , ( x ) - g , ( z ) [ is bounded a w a y / r o m zero uni/ormly on compact subsets of ~ a _ {z}.
(v) I f (~ is any hyperbolic line in ]]a ending at z then there is a relatively compact subset K o / B a and a sequence of distinct elements gn in G such that g,(a) N K 4 0 . (vi) For some (or all) x in B 3 there is a Stolz region T at z and a sequence of distinct
elements g~ in G such that g~(x)--->z in T.
I f h is n o w a MSbius t r a n s f o r m a t i o n w h i c h m a p s B a o n t o H a, t h e n hGh -1 a c t s o n H a a n d C a n d so m a y be r e g a r d e d as a g r o u p of m a t r i c e s . T h e p o i n t s of a p p r o x i m a t i o n of hGh -1 are t h e i m a g e s u n d e r h of t h e p o i n t s of a p p r o x i m a t i o n of G a n d T h e o r e m s (1)(v) shows t h a t t h i s d e f i n i t i o n is c o n j u g a t i o n i n v a r i a n t a n d so is i n d e p e n d e n t of h.
I n t h e special case w h e n A(G) is a p r o p e r s u b s e t of S ~ we c a n choose h so t h a t c~ qA(hGh-1). I n t h i s case we let a be t h e v e r t i c a l line t h r o u g h z o n C a n d we c o n c l u d e t h a t z is a p o i n t of a p p r o x i m a t i o n if a n d o n l y if t h e r e is a p o s i t i v e c o n s t a n t k w i t h
k for i n f i n i t e l y m a n y g in hGh -1.
L I M I T P O I N T S O F K L E I N I A N G R O U P S 5 W e n o w l e t
hOh -1
= {g,} w h e r eg ' = c~ d g a ~ d n - bncn= 1
a n d we h a v e p r o v e d t h e following result.P R O P O S I T I O N l . I n
the above situation z is a point o/ approximation o/ hGh -1 i/
and only i/there is a positive number k such that
klc,,I
/or inlinitely many gn in hGh -1.
P R O P O S I T I O N
2. 17/Z is a/ixedpoint o/the loxodromic element gEG, then z is a point o]
approximation.
Proo/.
W e can a s s u m e w i t h o u t loss of g e n e r a l i t y t h a t z is t h e a t t r a c t i v e f i x e d p o i n t . T h e n for e v e r y x E ~ ( G ) ,g-h(x)
c o n v e r g e s t o t h e o t h e r f i x e d p o i n t .T h e p a r a b o l i c case is s o m e w h a t m o r e c o m p l i c a t e d . W e n o r m a l i z e G so t h a t i t a c t s on 1t 3 a n d so t h a t z-~ z + 1 e G. L e t J b e t h e s t a b i l i t y s u b g r o u p of ~ ; i.e.,
J = {g e Gig ( ~ ) = ~ }.
W e r e c a l l t h a t i n general, if we h a v e a d i s c r e t e g r o u p G a c t i n g on, s a y H 8, a n d a s u b g r o u p J c G, t h e n t h e set A c t I 3 is
precisely invariant under J
if for e v e r y g E G e i t h e r(i)
g e J
a n dg ( A ) = A ,
o r (ii)g C J
a n dg ( A ) N A
= 0 .I t is well k n o w n (see, for e x a m p l e , L e u t b e e h e r [9] o r K r a [8, p. 58]) t h a t if
z ~ z + l E G , t h e n f o r e v e r y g = ( : bd) w h i c h i s i n O b u t n o t i n J , Icl>~l. A s a n i m m e d i a t e
c o n s e q u e n c e of this, we o b t a i nL~MMA 1. Let z-->z + l be an element o/the discrete group G acting on H a. Then A = { ( z , x ) E H a l x > l }
is precisely invariant under J, the stability subgroup o/ co.
W e conclude t h a t n o o r b i t can a p p r o a c h oo i n a S t o l t z r e g i o n a t oo a n d so we h a v e p r o v e n P R O P O S I T I O N
3. I / Z is the /ixed point o/ a Tarabolic element o/ G, then z is not a point ol approximation.
w
I n t h i s s e c t i o n we e x p l o r e t h e r e l a t i o n s h i p b e t w e e n p o i n t s of a p p r o x i m a t i o n a n d f i n i t e - s i d e d f u n d a m e n t a l p o l y h e d r a .
W e n e e d a d e f i n i t i o n of f u n d a m e n t a l p o l y h e d r o n w h e n t h e r e a r e n o t n e c e s s a r i l y f i n i t e l y m a n y sides. I n t h i s p a p e r , we r e s t r i c t ourselves t o
convex
p o l y h e d r a .A. F. B E A R D O N A N D B. M A S K I T
A (convex)
polyhedron P
is an open subset of B 3 (or of I t 3) defined as the intersection of countably m a n y half-spaces Q~ with the following property. Each Q~ is bounded b y a hyperplane Si; the intersection of Si with P , the closure of P in B 3 is called aside
of P.We require t h a t a n y compact subset of B a meets only finitely m a n y of the S~: then the b o u n d a r y of P in B 8 consists only of sides.
The polyhedron P is a (convex)
[undamental polyhedron
for the discrete group G if (a) no two points of P are equivalent under G.(b) E v e r y point of B a is equivalent under G to some point of P . (c) The sides of P are pair-wise identified b y elements of G.
(d) E v e r y x in B 3 has a neighbourhood t h a t meets only finitely m a n y translates of P.
We r e m a r k t h a t there is a Fuchsian group and a polygon P which satisfies (a) and (b), b u t not (c). F o r Fuchsian groups (d) is a consequence of (a), (b) a n d (c).
P R O P O S I T I O N
4. A point o/ approximation z o/ G cannot lie on the boundary o/ a convex/undamental polyhedron Po o/G.
Proo/.
As P0 is convex we can select a hyperbolic line a joining a point x in P0 to the point of approximation z. Theorem 1 (v) is applicable and this is in direct contradiction with the defining p r o p e r t y (d) of P0.One easily sees t h a t the identification of sides of P induces an equivalence relation on P , each equivalence class containing only finitely m a n y points.
I t is well known t h a t there is at least one convex f u n d a m e n t a l polyhedron for every discrete group. A particularly well known example is the
Dirichlet/undamental
polyhedron P0 formed as follows: We start with say 0 E B a where 0 is not fixed b y a n y element of G.For each non-trivial g E G, we form
Qg = {yEBaIQ(y, 0)<~(y, g(0))}.
One easily sees t h a t Qg is a half-space, and t h a t P0 = NgQg is a f u n d a m e n t a l polyhedron for G.
For a n y polyhedron p c B a, p is the relative closure of P in Ba; we let P* be the intersection of S * with the closure of P in ~a.
Our n e x t definition is concerned with parabolic fixed points; t h e y are limit points b u t t h e y m a y have aspects similar to ordinary points. We assume t h a t
z EC
is fixed point of some parabolic element of G, and let J be the stability subgroup of z. J is t h e n a Kleinian group with exactly one fixed point; all such groups are known (see Ford [4], p. 139). I n order to examine the possibilities, we assume t h a t G acts on H a, and t h a t z = c~.LIMIT PO~TS OF ~J~V.IN~ GROUPS 7 A cusped region U is a s u b s e t of C with t h e following properties. U is precisely i n v a r i a n t u n d e r J , a n d U is t h e u n i o n of t w o disjoint n o n - e m p t y o p e n half-planes.
One easily sees t h a t a cusped region U can exist o n l y if J is a finite extension of a cyclic group, a n d in this case U N A(G) = 0 . W e s a y t h a t z is a cusped parabolic fixed p o i n t if either there is a n o n - e m p t y cusped region U, or if J is n o t a finite extension of a cyclic group.
T h e existence of parabolic fixed points which are n o t cusped is given in Maskit [12].
TH~OR]~M 2. I] there is a convex/undamental polyhedron P / o r G with/initely many sides, then every limit point o/ G is either a point of approximation or is a cusped parabolic fixed point.
Proo/. W e start with the well k n o w n fact t h a t e v e r y p o i n t of P* is either in ~(G) or is a cusped parabolic fixed point. U n f o r t u n a t e l y , there is no r e a d y reference for this fact, a n d so we outline a proof here.
T h e identifications of the sides of P induce an equivalence relation on P , a n d on P*.
F o r each p o i n t z EP*, the set of points equivalent to z is called the (unordered) cycle at z.
Since P has finitely m a n y sides, t h e cycle contains finitely m a n y points.
W e n o w consider z in P* a n d conjugate so t h a t z = oo a n d t h e elements of G a c t on H 8.
W e choose gl ... gr in G so t h a t t h e cycle of co on P* is (g0(w), g~l(oo) ... grl(OO)} where, for convenience, go is t h e identity.
N o w let J be t h e stabilizer of co in G a n d J0 t h e s u b g r o u p of parabolic elements (and go) t h a t fix ~ (J0 m a y be trivial). I f ~ eg(P*) where gEG we can c o n s t r u c t a geodesic a f r o m a p o i n t in g(P) t o ~ . This implies t h a t for some i, 0 <~i <r, g~g-l(a) is a geodesic e n d i n g at ~ a n d so g~g-lEJ. W e conclude t h a t
J EJ U Jgi U ... U Jg~.
B y Propositions 2 a n d 4, J c a n c o n t a i n o n l y elliptic a n d parabolic elements a n d we see f r o m [4, p. 140-141] t h a t in this case there are elliptic elements e~ ... e~ such t h a t
J = Jo U Joel ... U Joe s.
W e conclude t h a t g lies in one of a finite n u m b e r of cosets John, h~EG.
I f J0 is trivial, t h e n a n e i g h b o u r h o o d of oo in H a U C meets only a finite n u m b e r of images of P a n d so ~
e~(G).
I f J0 is n o t a cyclic group, t h e n b y definition, ~ is a cusped parabolic point.
F i n a l l y if J0 is cyclic t h e images of P lie u n d e r one of t h e finite n u m b e r of euclidean c u r v e d sides of P or t h e h~(P) or are translations u n d e r J0 of these images a n d so a cusped region exists in this case.
A . F . B E A R D O N A N D B. M A S K I T
We now assume without loss of generality, t h a t 0 E P . Let zES 2, and let a be the line from 0 to z. If a intersects only finitely m a n y translates of sides of P, then for some gEG, g(z)EP*, and so by the above remark either zE~(G) or z is a cusped parabolic fixed point. Observe t h a t this situation must arise if zE~(G), for the euclidean diameter of translates of P must converge to 0.
The only possibility left is t h a t a passes through infinitely m a n y translates of some side M and in this case zEA(G). Then there is a sequence {gn} of distinct elements of G, and there is a sequence of points {Yn} on M, so t h a t g~(a) N M = {y~}. We can assume t h a t y=-~y. If y E B 3, then b y Theorem 1 (v) z is a point of approximation. If, as we now assume y CB 3, then b y the remarks above, y is a cusped parabolic fixed point. We again change normalization so t h a t y = ~ , and we let J be the stability subgroup of ~ .
If J is not a finite extension of a cyclic group, then there is a compact set K c C, so t h a t for every z'EC, there is a j E J with j(z')EK. Hence, we can choose a sequence {?n) of elements of J so t h a t j~ogn(z)EK, and j ' ~ o g n ( 0 ) - ~ . Observe t h a t this latter condition implies t h a t infinitely m a n y of the {inog=} are distinct.
If J is a finite extension of a cyclic group, then we can assume t h a t z ~ z + l E J , the cusped region is V = {z [ lira z ] ~> t}, and t h a t no translates of z lies in V. E x a c t l y as above, we can find a sequence {~} of elements of J so t h a t
lira 0"~og~(z))] < t, IRe (i~og~(z))] < 89
This concludes the proof of Theorem 2 as we have now verified Theorem 1 (iii).
We remark first t h a t as a corollary to the proof, we have the following well known statement.
COROLLARY 1. Let P be a convex [inite sided polyimtron /or G. Let p*o be the relative interior o/ P*. Then no two points o/ p*o are equivalent under G, and every point o/
~(G) N S 2 is equivalent under G to some point in the closure of p*o.
For the following applications we recall t h a t G is elementary if A(G) is a finite set.
COROLLARY 2. Let G be non-elementary. Then the set o/points o/approximation has positive Hausdor// dimension.
Proo/. I t was remarked b y Myrberg [13] t h a t every non-elementary discrete group q contains a Schottky subgroup G1, defined by say 2n circles. G 1 is then a discrete group of the second kind, with a finite-sided fundamental polyhedron. I t was shown b y Beardon [2] t h a t for every such G1, A(G1) has positive Hausdorff dimension. Since G 1 is purely loxodromic, A(G1) contains only points of approximation for G1, and so for G.
L I M I T POLNTS OF K L E I N I A N G R O U P S
COROLLARY 3. Let G have a finite-sided /undamental polyhedron, then the points o/
approximation o/A(G) are uniformly approximable, i.e., there is a constant k > 0 so that,/or every point of approximation z, there is a sequence (gn} of distinct elements of G with
I <kR .
Proof. L e t Pl ... Pr be the parabolic vertices on P . I n the notation of the proof of Theorem 1 we find t h a t if yn-~y, y =pj, then jnog,(O) remains outside some neighbourhood of the set (jnogn(z)}. I f we consider G as now acting in B a this means t h a t (retaining the same notation despite conjugation),
]hog~(z) -i~ogn(0) l >/k The result now follows b y (1) and (2).
A corollary of the above is the following theorem of Ahlfors [1].
COROLLARY 4. Let G have a finite sided ]undamental polyhedron. Then the 2-dimen- sional measure o / A ( G ) is either zero or 4~.
Proof. The proof is essentially immediate from Corollary 3, and the fact r e m a r k e d above, t h a t if G is of the second kind, then
R$<
geG
E x a c t l y the same considerations yield
COROLLARY 5. I f G has a finite-sided fundamental polyhedron, and if
<
geG
then the t-dimensional measure of A(G) is zero.
w
I n this section we prove the converse of Theorem 2. Specifically, our goal is to prove.
THEOREM 3. Let P be a convex/undamental polyhedron/or the discrete group G, where every point o/A(G) either is a point o/approximation or is a eueped parabolic fixed point.
Then P has finitely many sides.
Proof. Throughout we assume t h a t P is a convex f u n d a m e n t a l polyhedron for the discrete group G which, for the m o m e n t , is assumed to act on B a. I f P has infinitely m a n y sides, these accumulate at some point z on P . We begin b y showing t h a t zEA(G).
10 A . F . B E A R D O N A N D B. M A S K I T
L E M M A 2.
Let M1, M~, Ms, M~ be sides o/ P where there are pairing trans/ormations gl, g2EG with g~(M,)=M~. Then, gl=g~ i~ and only i / M I = M ~.
Proo[.
L e t S1, S 1' be t h e h y p e r p l a n e s on which M1, M I ' , respectively, lie, a n d letQ1, QI'
be t h e half spaces which are b o u n d e d b y $1,$1",
respectively, a n d which contain P . I f M S does n o t lie on S1, t h e n M s c Q1, a n d gl(M~) NQ1 '= 0.
W e conclude t h a t gl(M~) can be a side of P o n l y ifM2cS1;
i.e.,M ~ = M 1.
This l e m m a shows t h a t infinitely m a n y distinct images of P a c c u m u l a t e a t z. As P is c o n v e x a n d locally finite t h e euclidean d i a m e t e r of t h e images of P u n d e r G converge to zero, t h u s zEA(G).
Proposition 4 t o g e t h e r w i t h t h e h y p o t h e s e s of t h e t h e o r e m n o w i m p l y t h a t z is necessarily a cusped parabolic fix-point. W e complete the proof b y showing t h a t this is inconsistent w i t h t h e a s s u m p t i o n t h a t infinitely m a n y sides of P a c c u m u l a t e a t z.
W e shall a s s u m e t h a t G acts on H a a n d t h a t z = ~ . N o w let J be the stabilizer of co a n d J0 t h e s u b g r o u p of parabolic elements of J . W e m a y a s s u m e t h a t Jo contains z - + z + l : Jo is either cyclic or of r a n k 2.
W e will need t h e following r e m a r k a b o u t c o n v e x polyhedra.
LEMMA 3. Let (z~,
x~),
i = 1 ...n, be a finite set of points o/P. Let B be the Euclidean convex hull o/the points z I
...z~. Then
(i)
there is a
t > 0so that
{(z,x)EHa]zEB, x > t } c P , and
(ii)no two distinct points o/ B are equivalent under J.
Proo/.
Since J keeps each horosphere x = c o n s t a n t i n v a r i a n t , conclusion (ii) follows f r o m conclusion (i).Since P is c o n v e x a n d co EP*, if (z, x0) EP, t h e n so does (z, x) for e v e r y x > x 0. Con- clusion (i) n o w follows f r o m t h e f a c t t h a t if ~ is t h e n o n - E u c l i d e a n line f r o m (zl, xl) to (z2, x~), t h e n t h e projection of v o n t o t h e z-plane is t h e Euclidean line f r o m z 1 t o z 2.
This leads easily to
LV.MMA
4. Let z ~ - ~ in P with Zn=(un+iv~, X,).
(i)
I / Jo is cyclic, then v~ +x~ is unbounded.
(ii)
I / J o is o/rank 2, then u~ +v~ is bounded, x~ is unbounded.
Proo/. If
t h e conclusion of (i) fails then, b y L e m m a3,
P contains a subset of t h e f o r m [u', + oo) • Iv', v"] • [x', + co)(v'
< v ' ) a n d this contains points e q u i v a l e n t u n d e r J0. T h e proof of (ii) is similar.LIMIT P O I N T S O F K L E I N I A N G R O U P S 1 1
W e i m m e d i a t e l y deduce t h a t if z n is a sequence of distinct points in A(G)N P* t h e n z~+-~ co. I n d e e d in (i) we have
xn=O
a n d z~r U so Ivnl ~< V* whereas in (ii)x~=O.
T h e hypothesis of t h e T h e o r e m together with Proposition 4 n o w implies t h a t P* contains o n l y finitely m a n y limit points, in particular the cycle of ~ is finite.I f infinitely m a n y sides M n of P m e e t co we can select g~ in G where
g~(P)
a b u t s P along M~. B y L e m m a 2, these g~ are distinct. I t is evident t h a t P can a b u t at m o s t one other translate ofg(P)
u n d e r J0 a n d so we conclude t h a t t h e g~ lie in infinitely m a n y distinct cosetsJog.
This implies t h a t t h e set (g~l(co)} is an infinite subset of P c o n t r a r y to our previous remark. W e have p r o v e dLEMMA 5.
Only /initely many sides o / P pass through co.
W e have assumed there is an infinite sequence of sides
Mn
of P a c c u m u l a t i n g at co.T h e previous l e m m a implies t h a t we m a y assume t h a t n o n e of these contain ~ . W e select z n on M , with z~-~co a n d choose distinct g~ so t h a t
gn(P)
a b u t s P along M~.As co r M~ we conclude t h a t
gn(co) E C
a n d we can find a sequence ]~ in J0 with ?'~ o gn(co) lying in a c o m p a c t subset K of C. B y L e m m a 4 we observe t h a t j~(z~)-~co. I f ~ is t h e geodesic in]~og,(P)
joiningj~(zn)
to j~og~(~) we find t h a t t h e ~ m e e t a c o m p a c t subset of H 3 c o n t r a r y t o t h e a s s u m p t i o n t h a t the tesselation is locally finite. T h e proof is n o w complete.W e r e m a r k in closing t h a t we have used t h e fact t h a t we are dealing with 3-dimen- sional hyperbolic space in a crucial m a n n e r o n l y in the precise definition of cusped para- bolic fixed point. I n dimension 2, it is well-known, a n d one easily proves using L e m m a 1, t h a t e v e r y parabolic fixed p o i n t is cusped. I t is also well-known (see Greenberg [6] or M a r d e n [10]) t h a t a F u e h s i a n g r o u p has a finite sided f u n d a m e n t a l p o l y g o n if a n d o n l y if it is finitely generated. Combining these with the trivial fact t h a t a F u c h s i a n g r o u p has a finite sided f u n d a m e n t a l p o l y g o n if a n d only if as a Kleinian g r o u p it has a finite sided f u n d a m e n t a l polyhedron, we o b t a i n
COROLLARY
6. A Fuchsian group G is /initely-generated i/ and only i/
A(G)consists entirely o/points o/approximation and parabolic/ixed points.
References
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[4]. FORD, L. R., Automorphic /unctions, 2nd ed. Chelsea Publishing Co., New York, 1951.
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[7]. HEDLUND, G. A., Fuchsian groups a n d transitive horocycles. Duke Math. J., 2 (1936), 530-542.
[8]. KRA, I., Automorphic ]orms and Kleinian groups. W. A. B e n j a m i n Inc., Mass., 1972.
[9]. LEUTBECHER, A., U b e r Spitzen diskontinuierlicher Gruppen y o n lineargebrochenen Transformationen. Math. Zeitschr., 100 (1967), 183-200.
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Received June 6, 1973
