INEQUALITIES FOR CERTAIN FUCHSIAN GROUPS
B Y
A. F. B E A R D O N University of Cambridge, England
1. Introduction
I n one of the earliest papers on automorphic functions, Poincarg constructed func- tions automorphie with respect to a Fuchsian group b y means of the now well known Poincar@ series. If G is a Fuehsian group with oo an ordinary point of G, the convergence of the Poincar6 series depends upon the convergence of the series
E(G,~,t)= ~ Iv'(~)l ~
VeGwhere z is a n y ordinary point of G. I n 1882, Poincarg [15, p. 206] showed t h a t this series converges if t > 1.
Now suppose t h a t G is finitely generated. I f G is of the first kind, then [13, p. 181]
E(G,
z, 1) = + oo ( 1 . 1 )whereas if G is of the second kind, t h e n [13, p. 178]
~(G, z, 1) < -4- oo. (1.2)
An obvious question, then, is to w h a t extent can (1.2) be improved upon. I n this paper we show t h a t (I.2) is best possible when regarded as being a s t a t e m e n t applicable to all finitely generated Fuchsian groups of the second kind b u t nevertheless can be improved upon for a n y given group. More precisely, we prove the following two theorems.
T H ~ o ~ ]~ ~ 1. Given any number t satisfying t < 1, there exists a / i n i t e l y generated Fuch- sian group o] the second kind with oo an ordinary point o] G and with
~(G, z, t) = + oo ]or every ordinary point z.
222 A.F. BEARDO:N
T~wOR~M 2. Let G be a finitely generated _Fuehsian group o/the second kind with an ordinary point o/ G. Then there exists a real number t satis/ying t < 1 and
~(G, z, t ) < + ~ (1.3)
/or every ordinary point z.
Now let L be the set of limit points of G, denote b y re(L) the linear measure of L and again assume t h a t G is a finitely generated Fuchsian group. If G is of the first kind then obviously re(L) > 0 whereas if G is of the second kind, then re(L) = 0 [13, p. 324]. We prove t h a t this latter result is best possible when regarded as a statement applicable to all finitely generated Fuchsian groups of the second kind but can also be improved upon for a n y group. More precisely, we prove the following two theorems.
THEOREM 3. Given any number t satis/ying t < 1, there exists a finitely generated Fuch- sian group o / t h e second kind with co an ordinary point o/ G and with L having infinite t- dimensional Hausdor// measure.
TH~,OR~M 4. Let G be any finitely generated Fuchsian group o/the second kind. Then there exists a real number t satis/ying t < 1 such that L has zero t-dimensional Hausdor]/
measure.
The striking parallel between the first two and the last two theorems is explained b y the next result.
T H E O R ~ 5. Let G be a finitely generated Fuchsian group with c~ an ordinary point o/G.
1 / t is a real number such that
~ ( G , z, t) < + ~ (1.4)
]or some ordinary point z, then L has zero t-dimensional Hausdor]] measure.
We note immediately t h a t Theorem 1 is an immediate consequence of Theorems 3 and 5 and t h a t Theorem 4 is an immediate consequence of Theorems 2 and 5; thus we need only prove Theorems 2, 3 and 5. The proofs of Theorems 2 and 3 are long and for the benefit of the reader it seems desirable to discuss these results in a more general context before giving the proofs.
]First, we write mt(L ) for the t-dimensional Hausdorff measure of L and use d(L) to denote the Hausdorff dimension of L. This is defined b y
d(L) = inf {t > 0: mt(L ) = 0}
and the details of the construction of the measures m t can be found, for example, in [5].
Next, we write
I N E Q U A L I T I E S F O R C E R T A I N ] ~ U C I I S I A N G R O U P S 2 2 3
O(G) = inf {t>O: ~.(G, z, t ) < + ~ } (1.5) where z is an ordinary point of G. As is well known, O(G) is independent of z. An immediate consequence of Theorem 5 is the following (weaker) result.
C o ~ o r . L A ~ X .
In the above notation, d(L) <<.O(G).
The conclusion in Theorem 5 has been proved in the case when G is a S c h o t t k y group by Akaza [2], [3] and [4]. Our proof of Theorem 5 is, however, quite different.
Theorem 5 contains two well-known b u t non-trivial results. I f G is a finitely generated group of the first kind, then
ml(L )
> 0 and so using Theorem 5 we can deduce (1.1). I f G is a finitely generated group of the second kind, t h e n it is v e r y easy to establish (1.2) and so, using Theorem 5 again, we can deduce t h a t m l ( L ) = 0 .I n [9] Dalzell proved t h a t if G is a finitely generated Fuchsian group of the second kind and if G contains no parabolic elements, t h e n
IV'(z)l
log(IV'(z)1-1)
< + oo (1.6)V e G
for every ordinary point z. Theorem 2 is clearly an i m p r o v e m e n t of this result b o t h in t h a t (1.3) is stronger t h a n (1.6) and also t h a t G m a y contain parabolic elements.
The group Gz generated b y the elements.
P(z)=z+~, E ( z ) = - 1 / z ,
~ > 0 ,is called a tIecke group and is of the second kind if 2 > 2 . I n [7] the author studied the function (~(Gz) as a function of 2 (note t h a t the notation in [7] differs from t h a t used here;
the 6(G) used in [7] is twice t h a t defined b y (1.5)). I n particular, it was proved t h a t (in our present notation) 6(G~)> 89 t h a t
~(a~) = 89 +O(~t-1)
as ~ + co and t h a t (~(G~)<1 if ~ 2 . 8 .... The natural conjecture was then made t h a t 8 ( G z ) < l if 2 > 2 (that is, if Gx is of the second kind) and we see now from Theorem 2 t h a t this is so.
I n [6, p, 734] the author showed t h a t there exists a finitely generated Fuchsian group with
d(L)>
89 This is contained in the m u c h stronger Theorem 3 and indeed, Theorems 3 and 4 completely solve the problem of the range of values ofd(L)
in the case of Fuchsian groups.I n t h e last few years, several papers h a v e appeared in which there are estimates of
mt(L )
for various Fuchsian and Kleinian groups (e.g. [1], [2], [3], [4] and [6]). Some of thr results in this paper have been generalized so as to be applicable to Kleinian groups and so generalize some of these results. I t is hoped to publish these later.224 A.F. BEARDOI~
The theorems stated above are all concerned with finitely generated Fuchsian groups.
The results have been stated this w a y for brevity; the real requirement is the geometrical one t h a t the groups possess a fundamental region having a finite n u m b e r of sides and it is known t h a t these two conditions are equivalent (e.g. [11], [14]). Indeed, if G is finitely generated, the f u n d a m e n t a l region N0, defined as the set of points hyperbolically closer to a point w t h a n to a n y other image of w (see [13, p. 146]), has a finite n u m b e r of sides.
This follows from the results contained in [11]. We shall use these facts implicitly through- out this paper.
We can easily see t h a t Theorem 5 is false for infinitely generated Fuchsian groups and we give two counterexamples. First, it is easy to construct an infinitely generated Fuchsian group of the second kind with ml(L ) > 0 . To do this one simply constructs a se- quence of hyperbolic elements, each leaving the unit disc invariant and having the isometric circles of all of these elements and their inverses external to each other. This construction can be carried out in such a m a n n e r t h a t the images of ~ under these elements accumulate at a set of positive one-dimensional measure and so if G is the group generated b y these elements, G is of the second kind and so ~(G, z, t ) < + co. B y construction, however, ml(L ) > O.
A counterexample of a different t y p e is suggested b y a r e m a r k of Tsuji [17, p. 515].
Here Tsuji suggests the construction of an infinitely generated group of the first kind in which ~(G, z, 1) < + co and again, the existence of such a group shows t h a t Theorem 5 is false for infinitely generated groups.
The remainder of the p a p e r consists of the proofs of Theorems 2, 3 and 5. F r o m now on, and without further mention, we will reserve the symbol G to denote a finitely generated Fuchsian group and the symbol L for the set of limit points of G.
2. The proof of Theorem 5
L e t G be a group satisfying the hypotheses of Theorem 5. I f GI=AGA -1 for some bilinear transformation A satisfying A - ~ r L, then G 1 also satisfies the hypotheses of Theorem 5 and further, ~(G, z, t), ~(G1, Az, t) converge or diverge together. Also, the set of limit points of G 1 is A(L) and it is easily seen t h a t me(L) and m~(A(L)) are zero or positive together (this follows as A and A -1 satisfy a Lipschitz condition of order 1 on some neigh- bourhood of L and A(L) respectively). Thus we m a y consider G 1 rather t h a n G and this implies t h a t without loss of generality we m a y assume t h a t the unit circle {z: ]z I =1}
is the principal circle of G.
The proof of Theorem 5 depends on a theorem on Diophantine approximation for Fuchsian groups proved b y R a n k i n [16] and Lehner [13, p. 334]. The form of this result
I I ~ E Q U A L I T I E S F O R C E R T A I N I ~ U C I t S I A N G R O U P S 225 given in [13] is not in the form best suited to our needs and it is simpler to deduce a modified version directly from Lehner's generalization [13, p. 181] of a result of Hedlund [12, p. 538].
We need
Lv, M~A 2.1, [13, p. 181]. Let G be a finitely generated 2'uehsian group with {z: l z [ = 1}
as its principal circle. T h e n there exists a constant m satis[ying 0 < m < 1 and depending only on G with the ]ollowing property. I / ~ is a limit point o] G but not a parabolic vertex, then there exists a sequence o / p o i n t s z~ tending radially to ~ as n---> ~ and a sequence o/distinct elements V~ in G with ] V;l(z~) [ <m.
With ~, % and V~ as in Lemma 2.1, we have
I~- v.(o)l < Ir + l~,-v.(o)l = ( i - l~.l)+ l~,-v=(o)l
= ( 1 -
I v.(o)l)+ (Iv,(o)l- Ix, l)+ l~.-v,(o) I
~<
(1-I v,(o)l)+2l~,- v,(o)l.
In order to estimate these last two terms, we write
(2.1)
and note t h a t
v~(z) = an z + ~ c,~+a---~.' la, l ~_ le, i ' = l
(1-lv~(o)l)<l-lV~(O)l~=(l§ -~
(2,2)Also, if a is the straight line segment joining the origin to V;l(z~), then V~(a) has end- points V~(0) and z~ and so
fo Ilength (or) [[e.[ (1 - m ) ] -~.
Iz,~- V.(O)l<length [V~(a)]=
Iv'~(~)l.ldzl <lenl~infol ~- V;~(~)I~<
Using this together with (2.1) and (2.2) we find t h a t l ~ - v.(0) [ < 3(1 -m)-~l c.
I -~
for infinitely m a n y n. If we now write
Q(V) = {z: [ z - V(O)I <3(1 -m)-2[c[ -2}
where V (z) = az + 5
cz + 5 ' l a l 2 - Icl2= 1,
(2.3)
is in G, we see from (2.3) t h a t any limit point ~ of G t h a t is not a parabolic vertex lies in infinitely m a n y of the discs Q(V) for V in G.
226 .~. F. B E A R D O N
If (1.4) holds then t > 0 and
V e G , Vr ~ c ~
and so ~ [diam Q(V)]t< + oo.
V e G o V ~ - c ~
Thus for a n y positive e, we can find a finite subset K of G (including all those V in G for which V ~ = co) such t h a t
[diam Q(V)] t < e (2.4)
V e G - K
(where here and elsewhere the minus sign denotes the set-theoretic difference). If P denotes the set of parabolic vertices of G, then, as we have already seen,
L - 2 c U O(V)
V e G - K
for every finite subset K of G and this together with (2.4) implies t h a t m r ( L - P ) =0. As P is a countable set, ms(P)=0 for all t > 0 and so m~(L)=0 if (1.4) holds. The proof is now complete.
I n view of the fact t h a t this result is perhaps, the basic result of this'paper,rthe author feels t h a t it is worth giving a second, and completely different, proof of it. The above proof does not depend on the fact t h a t (1.1) holds for groups G of the first kind nor on the fact that ml(L ) =0 for groups of the second kind and so gives an alternative proof of these results. If we use the fact t h a t ~(G, z, t) converges if t > l and diverges for t = l when G is of the first kind we see t h a t Theorem 5 has been proved for groups of the first kind. We thus assume t h a t G satisfies the hypotheses of Theorem 5 and is of the second kind. The fundamental region N o constructed with z = 0 as its centre (by considering a conjugate group we m a y assume t h a t no element of G other than the identity fixes z=O) has a finite number of free sides s 1 ... Sn which we regard as open arcs of ]z] = 1. Clearly {z: ]z] = 1 } is the disjoint union
n
{.:[.I=~}=LuEu( U UV(s,))
V c G t ~ l
where L is the set of limit points of G and E is the set of end points of the free sides and their images. As there are only countably m a n y free sides, ml(E ) = O. We need the following result.
LEMMA 2.2, [8]. Let I be an open subset o/the interval J = [ 0 , 1] with m l ( I ) =1. I / t h e components In o / I have length an and i/ ~a~ converges /or some fl with 0 < fl <~ 1, then m Z ( J - 1 ) =0.
I N E Q U A L I T I E S F O R C E R T A I N F U O H S I A N G R O U P S 227 B y modifying this result so t h a t it applies to subsets of {z: Iz[ =1} rather t h a n J or b y considering a conjugate group to G so t h a t the limit set is contained in J , we see t h a t mt(L U E)=0 if
~ [length ( V(s~))] ~ < + oo. (2.5)
Y'eG i=l
As length (V(s~)) = fs IV' (z) l ldzl <- M lcl -~
(this holds as the orbit of Go cannot accumulate at a n y point in the closure of a n y s~) we see t h a t (1.4) implies (2.5) and hence t h a t mr(L)=0. This completes the second proof of Theorem 5.
3. The proof of Theorem 3
To prove Theorem 3 it is sufficient to consider a n y n u m b e r t satisfying 0 < t < 1 and to construct a group G with d(L) >~ t. The group t h a t we shall use is the Hecke group G[e]
generated b y the transformations
P(z) = z + 2 ( 1 +e), E(z) = - 1 / z (3.1) where s is a real, positive parameter. The limit set L of this group is an unbounded subset of the real line; thus Go is not an ordinary point of G[e]. Theorem 3 requires t h a t Go be an ordinary point of G and this condition is easily met. We shall show t h a t for sufficiently small 6, we have d(Lf] [ - 1 , 1])>~t. I f AG[e]A -1 is a n y conjugate group which has co as an ordinary point, t h e n
d(A(L)) >~ d(A(L f] [ - 1, 1])) >~ d(L fl [ - 1, 1]) >7 t,
the second inequality holding as A -1 satisfies a Lipschitz condition with exponent 1 in some neighbourhood of A(L f] [ - 1, 1]) (the results contained in the Appendix of [6] show t h a t the first two inequality signs could be replaced b y equality signs; we shall not need this however). I n a n y event, AG[e] A -1 is a finitely generated Fuchsian group of the second kind (G[s] is of the second kind) with Go as an ordinary point and with d(A(L))>~t. We therefore need only prove t h a t
d(L f] [ - 1, 1]) >t t (3.2)
where L is the set of limit points of G[s].
I t will be helpful to bear in mind during the proof t h a t the region {x+iy: Ixl < 1 + 6 , x ~ + y 2 > l }
228 i . F . B E A I ~ D O N
is a f u n d a m e n t a l r e g i o n for G[s] a n d t h a t t h e a c t i o n of E is a n i n v e r s i o n in t h e b o u n d a r y of t h e closed disc
Q = {z: Iz] < 1 } (3.3)
followed b y a r e f l e c t i o n i n t h e i m a g i n a r y axis. W e shall use Q t o d e n o t e t h e disc (3.3) t h r o u g h o u t t h i s p r o o f a n d w i t h o u t f u r t h e r m e n t i o n .
F o r e a c h n o n - z e r o i n t e g e r n, define
Vn(z) = Epn(z)
which is i n G[s], a n d , for e a c h f i n i t e sequence n 1 .. . . , nk of n o n - z e r o integers, define V(nl, ..., nk)(z) = V ~ , . . . V ~ k ( z )
(3.4)
a n d Q(nl, ..., nk) = V(n 1 . . . nk)
(Q). (3.5)
As V~(Q)c Q we see t h a t for a n y sequence n 1 ... nk,
QD Q(nl) ~ Q(ni, n2) D ... D Q(n I ... nk) a n d also t h a t if r ~ s , t h e n
Q(nl . . . nk, r) N Q(n 1 .. . . . nk, s) = O. (3.6) T h e s e r e s u l t s s h o w t h a t t h e s y s t e m of discs
{Q(nl ... n k ) : / c > 0 ; n 1 .. . . . n k ~:0}
y i e l d s a C a n t o r - l i k e c o n s t r u c t i o n w i t h r e s i d u a l set
L I = 5 U V(Q) (3.7)
k ~ l V~G~
where Gk is t h e set of e l e m e n t s of t h e f o r m (3.4) for v a r y i n g n~ . . . n~ b u t f i x e d k. W e s h a l l n e e d t h e following e l e m e n t a r y result.
L E ~ M A 3.1. L 1 is a subset o/ L N [ - 1 , 1].
Proo/. W e see f r o m (3.5), (3.6) a n d (3.7) t h a t t h e p o i n t s of L 1 a r e p r e c i s e l y t h o s e p o i n t s t h a t c a n be w r i t t e n in t h e f o r m
Q(nl . . . nk) (3.8)
for s o m e f i x e d i n f i n i t e sequence nl, n~ . . . As
I N E Q U A L I T I E S F O R C E R T A I N FUCHSIA:hi GROUI~S 229 V ( n . ..., n ~ ) ( ~ ) e Q ( n ~ , ..., n~_~) ~ Q
we see t h a t the point (3.8) of L I is in the closure of the orbit of ~ (which itself is in L) and hence is in LN [ - 1 , 1] provided t h a t
]Q(n 1 ... n~)] -> 0 (3.9)
as k-> ~ for every fixed sequence nl, n2, ... (here and elsewhere in this proof we use ]A[
to denote the diameter of a disc A). Using (3.4), an elementary computation shows t h a t for all non-zero n,
I v'(z)l
<(1 +2s) -1 on Q and soIQ(n~, ..., nk) l <(1 +2e)-~lQ(n~, ..., nk-~)] <2(1 + 2 s ) -~
the second inequality following Dom repeated applications of the preceding one. This establishes (3.9) and completes the proof of L e m m a 3.1.
The techniques for estimating the Hausdorff dimension of a set formed from a Cantor- like construction arc reasonably well developed (see, for example, [5]). There are, however, two m a j o r difficulties to overcome in applying these techniques to our construction. The first is t h a t in passing from one stage of the construction to the next, one replaces, say, Q(nl ... nk) b y infinitely m a n y (rather t h a n finitely many) Q(nl, ..., nk, nk+l). This dif- ficulty is overcome b y selecting only a finite n u m b e r of suitable Q(n~ ... nk, nk+~) at each stage and using only these in the construction. The second difficulty is t h a t the ratios
]Q(~...., n~, n~+~)l" ]Q(nl ... n~)1-1
arc n o t well-behaved in the sense of constructions of this nature. We avoid this difficulty b y modifying the above construction of L 1 so as to avoid images of Q under successive applications of V1 or of V-1 (for it is these t h a t give rise to the badly-behaved ratios).
Roughly speaking, we replace V1 and V_I in G~ b y a set of elements V(11) ... V(1N), V(_1)1, T/(~) (to be described in detail later) and t h e n use the modified G 1 to generate a semi- group of transformations (each of which will still be of the form (3.4)). The images of Q under the transformations of the semi-group yield a Cantor-like construction with a residual set L 2 which is a subset of L 1. This and L e m m a 3.1 imply t h a t
d(L2) < d(L~ [ - l , 1]) < 1 (3.10) and then there remains a rather delicate estimation of d(L2) to show t h a t
lira lim sup d(L~) = 1 (3.11)
e-->0 N/>2
230 A.F. B E A I ~ D O N
which, together with (3.2) gives t h e required result. We proceed n o w with t h e formal proof.
L e t e be a positive n u m b e r a n d / V an integer satisfying N ~> 2 (e a n d N are t h e para- meters occurring in (3.11) a n d will be held fixed until just before t h e end of t h e proof).
Next, let G[s] be as previously described in (3.1) a n d let 1"i be t h e set consisting of t h e ele- m e n t s
(A) V2, V_~
... VN,V-N,
t o g e t h e r with t h e elements V(n i ... nk, m) where •1 . . . 9zk, m satisfy one of t h e following conditions:
(B) l<~k<~N,n~ . . . n z = l
and2<<.[m[<N,
(B') l < k < N , nl . . . n k = - I a n d 2 < l m I < N , (C) l <<.k<~N, n 1 . . . n k = l a n d m = - l , (C') l~<k~<2V, n i . . . n k = - - l a n d m = l .
W e shall refer to these elements as being of t y p e A, B, B', C a n d C' respectively.
H a v i n g defined 1"1, we n o w define 1"n for all positive integers n b y t h e i n d u c t i v e definition
1"n+1 =
{UV" UEPn,
V E l a l } = { U 1 ... U n + l : U i e P l , i = l . . . n - ~ - l } a n d further, defineL2=
N [.J V(Q).
(3.12)kffil Vr
I t is clear t h a t if V E Fk a n d T i a n d T 2 are in F i with T i # T2, t h e n V(Q) D VTI(Q)
a n d VTI(Q) N VT2(Q) = f~.
F r o m these facts a n d L e m m a 3.1 we can easily prove t h a t L 2 c L i c L N [ - 1 , 1]
a n d so (3.10) holds. I t remains therefore to establish (3.11).
To do this we need the concept of a spherical C a n t o r set. This is essentially a set con- s t r u c t e d in a similar m a n n e r to t h e classical Cantor set b u t with a little more metrical freedom in t h e construction. This c o n s t r u c t i o n m a y be carried o u t (as in our case) in t h e plane using discs instead of intervals a n d details of such sets t o g e t h e r with estimates of their H a u s d o r f f dimension can be f o u n d in [5].
INEQUALITIES FOR CERTAIN FUCHSIAN GROUPS 231
LEMMA 3.2. I n the above notation, L 2 is a spherical Cantor set constructed ]rom the discs {U(Q): u E r ~ , nJ>l}.
The construction of L 2 is t h a t of replacing U(Q), UEF~ b y U UV(Q)
VeFl
at each stage of the construction. We now write F = U nPn and so, b y virtue of L e m m a 3.2, we m a y rewrite [5, Theorem 4(if), p. 683] in our present notation to give the following result.
LE~tMA 3.3. / / 0 satis/ies 0 < 0 < 1 and i/
!UV(Q)l~ IU(Q)l ~ (3.13)
VeF1
/or all U in U, then d(L2) >~O.
The validity of L e m m a 3.3 thus depends upon L e m m a 3.2 which has y e t to be proved.
I n order to attain continuity of the basic ideas involved in the proof we proceed a little further before proving L e m m a 3.2. Our n e x t step is to establish the following simple result.
LEM•A 3.4. Let I~ be any integer greater than one and let the 19ositive numbers (51 .... , (~k, 6 and s satis/y 0 <(~j <~ < 1 and
0 ~<s ~<c$1 + ... +c~k < 1. (3.14)
Then 60 + . . . + 6~ >~ 1, (3.15)
where 0 = 1 - (1 - s ) ( 1 - 6 ) -1. (3.16)
We shall use L e m m a 3.4 b y taking the numbers (~1 .. . . . ~k to be the ratios
I UV(Q)I.]U(Q)]-I, UEF, V E F 1 (3.17)
for t h e n the inequality (3.13) is precisely (3.15). We thus obtain the estimate of d(L2) given b y (3.16) and L e m m a 3.3 if we establish L e m m a s 3.2 and 3.4 and verify t h a t with the above choice of c$1 ... ~k, the hypotheses of L e m m a 3.4 are satisfied. We now begin the t a s k of establishing these results.
Proo/ o~ Lemma 3.4. L e t v be the unique positive n u m b e r satisfying
~ + . . . + ~ - 1, (3.18)
thus O < v < l . Next,
232 A.F. B ~ D O X
f
1-6~ -~= d(x ~-') = ( l - v ) x-~dx>~ ( l - v ) (1-~j)>~ ( l - v ) ( 1 - 5 ) .
i i
Using this inequality together with (3.14) and (3.18) we have
k
1-s~> ~ ~(1-~-~)>~ (1-~)(1-5)
t = 1
a n d so using (3.16) we can easily deduce t h a t 0 ~<v. This together with (3.18) yields (3.15) and the proof of L e m m a 3.4 is complete.
Proo/o] Lemma 3.2. The definitions in [5, p. 680] i m p l y t h a t we m u s t establish the existence of positive constants A 1 and A 2 such t h a t (i) for all U in P and all V in Pl,
[ UV(Q)[ >~ AII U(Q)]
and (ii) for all U in F and all distinct T 1 and T 2 in F1,
(3.19)
where @ is defined b y
@[UTI(Q), UT~(Q)] >~A2] U(Q) I
dE, F) =inf {le-/[: eeE,/OF}.
(3.20)
W i t h our choice (3.17) of the 51 ... ~k in L e m m a 3.4 the constants s and ~ in L e m m a 3.4 also become bounds on the ratios (3.17) and so at this point it is advantageous to derive a general distortion theorem for the family U. An application of K o e b e ' s distortion theorem [10, p. 175] would give a short proof of L e m m a 3.2 b u t does not, however, seem strong enough to yield useful estimates for the constants appearing in L e m m a 3.4. We prove the following result in which the estimates are more explicit.
L~MMA 3.5. Let J = [ - 1 , 1], let I be any sub-interval o / J and let UEP. Then
(1/5) IX[ < I V(I)l-< 1~2~1 ~ (5/4) Ill.
Also, ~/ v e r l them 1UV(J)l < (5/6) 1U(J)[. (3.21)
We r e m a r k t h a t we are using I1] to denote the length of I (a one-dimensional disc).
No ambiguity will arise from the two uses of this symbol; indeed as Q has its centre on the real line and as elements of P leave the real line invariant, we do have
1U(Q)[ = I U(J) I" (3.22)
The proof of L c m m a 3.2 is easily completed. B y taking I to be the intersection of V(Q) with the real axis we have from (3.22) and L e m m a 3.5 t h a t
I N E Q U A L I T I E S F O R C E R T A I N F U C t I S I A N G R O U P S 233 I UV(Q)[ = I U(-/)[ ~ A l l U(Q)[
where A1 = (1/5) rain {I V(Q)[ : VEF1}
a n d this is positive as F1 is finite. This establishes (3.19). T h e proof of (3.20) is similar.
T h e set
( - 1 , 1 ) - U V(Q)
Vel~z
consists of a finite n u m b e r of open arcs a~ and, if T 1 a n d T 2 are distinct elements of F1, t h e r e exists a s u b a r c a of J lying b e t w e e n T~(Q) a n d T2(Q) with la[ ~>min ]as] > 0 . As U(a) lies b e t w e e n UTI(Q) a n d UT2(Q ) we m a y use L e m m a 3.5 a n d (3.22) to deduce t h a t
e[UT~(Q), UT~(Q)] ~> ] U(a)] ~> (1/5)]a[ 9 ] U(Q)[ >~ (1/5)[ U(Q)] (rain [a,])
which established (3.20). This completes t h e proof of L e m m a 3.2 subject to L e m m a 3.5.
Indeed, t h e proof of L e m m a 3.5 is the only o u t s t a n d i n g i t e m in our p r o g r a m m e so far, P r o o / o / L e m m a 3.5. L e t I , J a n d U be as in the s t a t e m e n t of L e m m a 3.5 a n d p u t w = U - I ( ~ ) (thus w is a real n u m b e r ) . Our first t a s k is to e s t i m a t e w. As U E P, we see t h a t
U = U~ ... U k = V ( n ~ ... n~, n~+~)
(U~EG)
where U k = V(n ... , ns+l), say. I f Uk is of t y p e A, t h e n r = s + 1 a n d [ n~+l I ~> 2. I f Uk is of t y p e /~ or B', t h e n r<~s a n d again, [n~+ll ~>2. Finally, if U k is of t y p e C or C', t h e n r<~s a n d either n ~ = l n ~ + ~ = - 1 , or n ~ = - 1 , n ~ + l - 1 . N o t i n g t h a t
w = (EPic... E P ~.+')-1( o o ) = P-~+~ ( E P - ~ . . . E P - ~ ) (0) which belongs to P-~,+~Q( - ns . . . - nl), we find t h a t
w E P - ~+~ Q( - n~). (3.23)
I f U k is of t y p e A, B or B ' , t h e n ] n~+ 1 ] ~ 2 a n d so replacing (3.23) b y t h e w e a k e r s t a t e m e n t : wEP-~+I(Q)
we find t h a t [w]/>3. I f Uk is of t y p e C or C', t h e n (3.23) becomes w EP(Q( - 1)) U P-~(Q(1))
P-~EP(w) EJ or P E p - I ( w ) EJ.
or, e q u i v a l e n t l y
This in t u r n implies t h a t
]w I >~ ~+(1+~)-1 >7/3
1 6 - 712907 A c t a mathematlca 127. I m p r l m 6 le 8 0 c t o b r e 1971
234 A. ~. B ~ D O N
where 2 = 2 + 2 s ~ > 2 . I n all case, then, we h a v e lwl ~>7/3. W e n o w p u t I = [ ~ , fl] (as we are only concerned with ] I I we m a y assume t h a t I is closed) a n d note t h a t
_ , -1 _ w)-2dx ]-1
,U([)] (f [Vt(x)[dx) (fl[ v (x)]dx) =(f (x-w)-2dx) (;l(x
IU(J) I
/
1 {((w+l)(w-1;}
= 1II
U s i n g - 1 ~ < f l ~ < l , we find t h a t if w>~7/3, t h e n
1 t (I)i<_ l l(w§
which gives t h e required estimate as w ~> 7/3. A similar a r g u m e n t establishes t h e result if w ~ < - 7 / 3 a n d this completes t h e proof of t h e first p a r t of L e m m a 3.5.
Finally, if VCF1, t h e n V = V(n I ... n~) for some n 1 .... , n~ a n d we h a v e V(J) ~ V(n~) (J).
A n e l e m e n t a r y c o m p u t a t i o n shows t h a t
] V(n~)(g)]
= ( n l ~ - 1 ) - 1 - ( n l ~ -~ 1 ) - 1 < 2/3as l n ~ [ ~>2. A p p l y i n g t h e first inequality in L e m m a 3.5 with 1 = V(J) we find t h a t ] UV(J)] <~ (5/4)(2/3) I U(J)[
a n d t h e proof of L e m m a 3.5 is complete.
L e m m a s 3.2 to 3.5 inclusive h a v e n o w been p r o v e d and, with t h e choice (3.17) of t h e 81, .-., 6k in L e m m a 3.4, we can use L e m m a 3.5 to o b t a i n estimates of t h e constants s a n d occurring in L e m m a 3.4. Indeed, t h e inequality (3.21) in L e m m a 3.5 shows directly t h a t we can t a k e (~ to be 5/6. Next, we note t h a t
~ + ... +(~= ~ Iuv(J)l. Iv(J)l -~. (3.24)
Ve F
I f we n o w write F = ( - 1, 1) - (J V(J) (3.25)
VeF~
t h e n F is an o p e n subset of ( - 1 , 1) and we have
]U(J) I = m~ (U(F)) § ~ I UV(J)[. (3.26)
VeF~
I N E Q U A L I T I E S F O R C E R T A I N F U C H S I A N GROUI~S 2 3 5
As Y is a u n i o n of open intervals, we deduce f r o m L e m m a 3.5 t h a t m l ( U ( F ) ) ~
(5/4)ml(F)[u(g)]
a n d using this inequality with (3.24) a n d (3.26) we find t h a t 31 + ... § = 1 - m I ( U ( F ) ) ] U(J)] -~ >~ 1 - ( 5 / 4 ) m ~ ( F ) .W e h a v e t a k e n 5 to be 5/6; we n o w define s b y
8 = 1 - ( 5 / 4 ) m l ( F ) .
L e m m a s 3.3 a n d 3.4 t o g e t h e r with (3.17) a n d (3.24) enable us to deduce t h a t d(L2) >~0 where 0 ~ 1 - ( 1 5 / 2 ) m l ( F ) . This gives
d(L2) >~ 1 - 8m~(F). (3.27)
Recalling t h a t in order to prove T h e o r e m 3 it is only necessary to establish (3.11), we n o w find t h a t it is only necessary to prove t h a t
lim lira sup m 1 (F) = 0
e-->0 N--~ c~
where F is defined b y (3.25). This is geometrically obvious; however, we prefer an analytic proof. To achieve this, we define a set T b y
N
T = I - (J V(n) (J) (3.28)
] n l = l
where I = ( - 1 , 1) a n d also, for convenience, define u ~ = l a n d v ~ = - 1 for each positive.
integer n. W e t h e n h a v e
N
F - T = [J V ( n ) ( J ) - (J V ( J ) = (J V ( n ) ( J ) - [J V ( J )
I n l ~ 1 V e F 1 n - - 1 , 1 V e F I
= [ F ~ V(1) (J)] U [ F N V ( - 1) (J)] (3.29~
as for every subset K of I , we h a v e
K - U V ( J ) = K n F .
V e F ~
Next, we h a v e
F (~ V ( u 1 . . . u~) (J) - V ( u 1 .. . . . u~) (T)
= V ( u l . . . u~) (J) - [J V ( J ) - V ( u 1 .. . . . u~) (T) = V ( u 1 .. . . . Ur) ( J - T ) - (J V(J)~
Vel~l VeF~
N
= [V(u~ . . . u~) ( IJ V(n) (J))] U [ V ( u x . . . ur) { - 1, + 1}] - LJ V ( J )
[ n [ ~ l VeFx
= V ( u 1 .. . . . u~, ur+l) (J) (J V ( u l . . . . , u~) { - 1, + 1},
236 A.F. B E A R D O N
t h e p e n u l t i m a t e equality following f r o m (3.28) a n d J = [ - 1, 1] and t h e last equality f r o m t h e definition of Pl. This gives
m l [ F (I V(ul, ..., ur) (J)] = m l [ V ( u l . . . ur) (T)] ~ - 7 / / , l [ V ( u I . . . $ r 4 1 ) (J)].
A similar e q u a t i o n holds with u 1 ... ur+l replaced b y v 1 ... V~+l a n d using these t w o equations for r = l .... , N - 1 a n d also (3.29) we find t h a t
N - 1 N - 1
m 1 (F) = m l (T) + ~ m 1 [ V ( u l . . . u~) (T)] + ~ m 1 [V(v 1 . . . Vr) (T)]
r = l r = l
+ m l [ V ( u 1 . . . uu+l) (J)] + m 1 [V(vl . . . VN+I) (J)].
As b o t h T a n d J are s y m m e t r i c a l with respect to tile i m a g i n a r y axis, we find t h a t m l [ V ( u 1 .. . . , Ur)(T)] = m l [ V ( v I .... , vr)(T)] a n d also t h a t a similar e q u a t i o n holds with T replaced b y J . Thus we h a v e
N - 1
m 1 (.F) <~ m 1 (T) + 2 ~ m 1 [ V(uz . . . u~) (T)] + 2 m 1 [ V(u 1 . . . U N + 1) ( J ) ] . (3.30) A l t h o u g h it is easy to obtain simple estimates for these terms it does n o t seem a trivial m a t t e r t o o b t a i n estimates delicate enough t o give the required information when e tends t o zero a n d N tends to oo.
We first estimate re(T). To do this note t h a t T consists of t h e origin together with t h e images u n d e r an inversion in the b o u n d a r y of Q of t h e intervals [ 2 r + l , 2 ( r + l ) - l ] r = - N, ..., N - 1 a n d the intervals ( - c o - [2(2 + 1]), (N2 + 1, + oo), where, as before, 2 = 2 + 2 e . This enables us to c o m p u t e ml(T):
2 N - I f 1 1 ] J ~z-1 2 e
m l ( T ) - N ~ + l + 2 ~ = o l ] ~ r - + l 2 ( r + l ) - l I ~ - ( 1 / N ) + 2 ~ - o ( ~ r ) e < ( 1 / N ) + 3 s " (3.31) W e n e x t estimate ml[V(ul .... , ur)(T)]. I f we p u t
arz + b~ ard~ - b~ cr = 1 (3.32)
V(u~ . . . u~) (z) crz + d /
we k n o w t h a t t h e pole of V ( u 1 .. . . . Ur) lies in p - l ( Q ) a n d so ]d~ ] > ]Cr I" T h u s we can o b t a i n t h e following estimate:
m l[V(u~ . . . ur) (T)] = f T l V ( u 1 .. . . . u~)' (z)[.
]a r
= f r l C r Z + d r ] - 2 1 d z ] < ~ ( I d ~ ] - I c r ] ) - e m ~ ( T ) . (3.33)
I ~ E Q U A L I T I E S x~og C E R T A I ~ F U C H S I A ~ G R O U P S 2 3 7
Our n e x t t a s k is to c o m p u t e Cr a n d dr. B y definition we h a v e V(u:t ... ur) = ( E P ) T a n d so if at,
br,
C r a n d dr are as in (3.32), we h a v eV ( u l . . . . , Ur+l) = V ( u l . . . ur) E P
(ar+l br+l --
\cr+1 d r + l ) = ( : : ~ : ) ( ~ 12)"
a n d s o
F r o m this we deduce t h a t cT+l=d ~ a n d d r + ~ = 2 d r - c r with initial conditions c1=1, d 1 = 2 . E l i m i n a t i n g cr a n d using s t a n d a r d techniques to solve t h e resulting difference e q u a t i o n (with c o n s t a n t coefficients) we find t h a t
cr = (pr_qr)(p_q)_~
~nd dr = (pr+l _qr+l) (p _ q ) - i
where p a n d q are t h e roots of x 2 - 2 x + l = O .
W e r e m a r k t h a t cr a n d dr are o n l y d e t e r m i n e d to w i t h i n a f a c t o r of - 1 (although dr/cr is unique) a n d this corresponds to t h e t w o choices of t h e ordered p a i r (p, q). I f we write
p = 89 q = 89
we find t h a t p > q > 0 a n d t h a t
(p -- q) ( [dr I - I or I ) = s _ 1) + qr(1 - q) ~> pr(p _ 1) (3.34) as .pq = 1 a n d p > g > 0 implies t h a t 0 < q < 1. F r o m this we can deduce t h a t
N - 1
Z < Z iv-q\2 2r=(v-q]2(v -l)
I f we n o w write /z = 89 2 - 4 ) I >~ (2s)~ (3.35)
we find t h a t p = l + s + # , q = l + s - # a n d so we h a v e
N - 1 4 # 2
2 2 - 4 ~< ( 4 s + 2 e 2) (2e)-~ ~< 3/e~
rZI (Idr l - Icrl)-2 < (s +/~)~(2 + c + #) 2~3
if r < 1, t h e p e m f l t i m a t e i n e q u a l i t y following f r o m (3.35). F r o m this a n d (3.33) we can deduce t h a t
N - 1
m l [ V ( u I . . . ur) (T)] ~< 3 m l ( T ) (~)-'~. (3.36)
r = l
N e x t , the e s t i m a t e (3.33) is valid with T replaced b y J a n d this a n d (3.14) yields
238 A . F . B E A R D O N
?~1 IV(u1, .... UN+I) (J)] < (P - q]2P-2m+l)ml(J) 9 \ p _ ] /
+ 3 Using (3.30), (3.31), (3.36) and (3.37) we find t h a t
ml (F) ~ ml (T) [1 + 6e-~J + 4 ( P - - q~ eP-2(N+ I) ~ (
(3.37)
~)(1 + 6 ~ ~) + 4 ( P - q]~P 2 ( ~ + 1 " ~ p - 1/
As p and q depend only on s and as p > 1, we deduce t h a t lim sup m 1 (F) ~< 3 e + 18 t/ss.
N ' - ~
As L (the limit set of G[e]) is independent of N, this inequality together with (3.10) and (3.27) implies t h a t
d(Ln [ - 1 , 1]) >~ 1 - 8 ( 3 ~ §189
I t is now clear t h a t if t < l , then for sufficiently small positive s, d(LN [--1, 1]) ~>t
and the proof of Theorem 3 is complete.
4. The proof of Theorem 2
We begin b y proving a lemma on Dirichlet series which will be used later and which does not depend on the notion of a Fuchsian group.
L]~MMA 4.1. Let A1, A S .... be a sequence o] positive numbers such that
n = l
converges i~ t > 89 I / a l , a S . . . . i8 any sequence o/ numbers satis/ying O <~an ~ A ~ ( n = l , 2 .... ) and i ] t satisfies 5/6 ~ t ~ 1, then
a~< ~ an + 6(1 - t) [A(2/3) + A(4/3)].
n = l n = l
Pro@ I f ] is defined b y
/ ( z ) = a s n - - 1
then / is defined and analytic on {Re (z) > 89 and satisfies
I N E Q U A L I T I E S F O R C E R T A I N F U C H S I ~ N G R O U P S 239
1/(~)1< ~ ~<A(x) (z=x+iy)
n ~ l
there. If [ x - 11 < 1/3, then A~ < A~ + A
(the inequality holding with one term of the sum according to whether A~ is not greater than or not less than 1) and so [/(z) I ~ A(~) + A(-~) on [ z - 11 < 89 Cauchy's inequality implies t h a t [/'(z)l ~6[A(~-)+A(~)] on ] z - I [ ~-~. Thus if ~<~t<~l, then/(t)<~/(1)+ I/(t)-/(1)l<~
](1) + 6 ( 1 - t ) [ A ( ~ ) +A(})] and this is the required result.
We return now to the theory of Fuchsian groups. As Theorem 2 is known (and easily proved) to be true when G has at most two limit points [7, p. 474] we assume t h a t G has uncountably m a n y limit points. As we have already mentioned in the proof of Theorem 5, it is sufficient to consider a conjugate group A G A -1 provided t h a t A - l ( ~ ) ~L, the set of limit points of G. Without loss of generality, then, we assume t h a t the elements of G preserve the upper half-plane and, of course, the extended real axis which we shall denote b y R 1. We note t h a t R l is considered as a subset of the extended complex plane and hence contains the single point at infinity.
As is well known, the upper half-plane can be given a hyperbolic metric and a normal (or Dirichlet) polygon N 0 constructed from this metric is a fundamental region for the action of G on the upper half-plane (for details of this, see [13, Chapter IV] where this is done for the disc rather t h a n the half-plane). We now wish to make certain justifiable assumptions on G and N 0. First, b y choosing the centre of No outside of some set of plane measure zero we m a y assume t h a t each parabolic cycle on the b o u n d a r y of N o consists of a single vertex and also t h a t (in the notation of [13, p. 149-151]) there are no accidental vertices of the first kind lying on R 1. We note t h a t as G is finitely generated, N 0 has only finitely m a n y sides ([11], [14]) and so every vertex of N o which lies on R 1 is either a parabolic vertex p or an accidental vertex q. Our choice of centre as given above implies t h a t in the former case the sides of N 0 t h a t meet at p are conjugated by a parabolic element of the group whereas in the second case, q is the intersection of a side of N o and a (closed) free side of N 0. This means t h a t q is the common end-point of two images of free sides of N 0 and t h a t some neighbourhood of q is covered b y the closure of the union of two images of N 0. These properties are preserved under conjugation; thus b y considering (if necessary) a conjugate group we m a y assume t h a t co or one of its images lies in the open set N 0.
After relabelling (i~ necessary) we m a y assume t h a t ~ 6 N 0. I t is more convenient in this proof to consider the action of G on the extended complex plane rather t h a n on the upper half-plane; thus we modify our notation and from now on denote b y N o the union of the normal polygon described above as No, its reflection in the real axis and its free sides on RL
~ : 0 A . F . B E A R D O N
T h u s N o is a f u n d a m e n t a l r e g i o n for t h e a c t i o n of G on t h e e x t e n d e d p l a n e ; i t is f i n i t e sided, s y m m e t r i c w i t h r e s p e c t t o t h e r e a l axis; i t c o n t a i n s a n e i g h b o u r h o o d of co a n d its v e r t i c e s h a v e t h e p r o p e r t i e s l i s t e d a b o v e . T h e free sides of N o are t h o s e of t h e N o as o r i g i n a l l y defined.
I n t h i s p r o o f we shall use A', .A a n d ~A t o d e n o t e t h e closure, t h e r e f l e c t i o n i n t h e r e a l axis a n d t h e b o u n d a r y r e s p e c t i v e l y of a set A a n d we shall use ] E I t o d e n o t e t h e l i n e a r m e a s u r e of a m e a s u r a b l e s u b s e t E of t h e r e a l line. W e shall also m a k e a n a t t e m p t t o a v o i d as m u c h as possible of t h e g e o m e t r i c a l a r g u m e n t t h a t is so c o m m o n in t h i s s u b j e c t . W i t h t h i s in m i n d we first c o n s t r u c t a f u n c t i o n z t h a t is d e f i n e d o n t h e e x t e n d e d p l a n e , t h a t satisfies a M i n i m u m P r i n c i p l e b o t h t h e r e a n d on t h e r e a l line a n d t h a t a n a l y t i c a l l y d e s c r i b e s t h e t e s s e l a t i o n of t h e p l a n e b y N o a n d its i m a g e s u n d e r G.
F i r s t , we s a y t h a t t w o e l e m e n t s U a n d V in G are a d j a c e n t if a n d o n l y if u(N~) n v(No) ~ ~).
N e x t , we p u t G O = {I} ( I is t h e i d e n t i t y e l e m e n t in G) a n d a s s u m i n g t h a t Go, G 1 .. . . . G~
h a v e b e e n d e f i n e d we define Gn+l as t h e set of t h o s e V in G s a t i s f y i n g (a) V C G O , G 1 .. . . . G~ a n d
(b) V is a d j a c e n t t o s o m e U i n Gn.
F r o m (a) we see t h a t t h e G~ are m u t u a l l y d i s j o i n t a n d so we c a n define a f u n c t i o n ~* o n
o0 ~o G
U n=0 Gn b y z*(V) = n if a n d o n l y if V E Gn. O u r i m m e d i a t e t a s k is t o show t h a t G = U n=0 n.
I f U a n d V a r e a d j a c e n t a n d if z*(U) is d e f i n e d a n d e q u a l t o n, say, t h e n (b) holds. I f (a) holds, t h e n b y definition, VEGn+I a n d so z * ( V ) = n + l w h e r e a s if (a) fails t o hold, t h e n
~*(V) is a l r e a d y d e f i n e d a n d is n o t g r e a t e r t h a n n. T h u s if U a n d V a r e a d j a c e n t , 7e*(U) a n d ~*(V) are e i t h e r b o t h d e f h l e d or b o t h u n d e f i n e d . I f V is n o w a n y e l e m e n t of G, t h e h y p e r - bolic line j o i n i n g co t o V(oo) crosses, in t u r n , t h e a d j a c e n t regions
No = 1(No), Vl(No), V2(No) ... V~(No) (V =: V~)
a n d as z * ( I ) is defined, so is ~*(V~). T h u s G = U n=O Gn a n d ~* is d e f i n e d on G. I f U is a d j a c e n t t o V, t h e n V is t o U a n d t h e a b o v e a r g u m e n t shows t h a t i n t h i s ease
[~*(u)-~*(v)] < l. (4.1)
W e also n o t e t h a t if P c o n j u g a t e s t h e sides of N o e n d i n g a t a p a r a b o l i c v e r t e x on ~N0, t h e n pn E G 1 for all n o n - z e r o i n t e g e r s n. A s i m i l a r s t a t e m e n t h o l d s for p a r a b o l i c v e r t i c e s on t h e b o u n d a r i e s of t h e i m a g e s of N o .
W e a r e n o w in a p o s i t i o n t o define t h e f u n c t i o n 7e m e n t i o n e d a b o v e . I f z E V(No) for s o m e V in G, we define ~(z) t o b e z*(V). T h i s defines ~ on a dense s u b s e t of t h e c o m p l e x p l a n e a n d we c o m p l e t e t h e d e f i n i t i o n b y t h e r e q u i r e m e n t t h a t
I N E Q U A L I T I E S F O R C E R T A I N F U C I I S I A I ~ G R O U P S 241
re(z) = lim inf re(w) (4.2)
/v-->z
the lower limit being t a k e n over w in U w e V(No).
T h e following r e m a r k s a n d l e m m a s describe some of t h e basic properties of t h e f u n c t i o n re. First, re only assumes t h e values + c~, 0, 1, 2, .... Next, a l t h o u g h (4.2) was only intro- duced t o define re on a nowhere dense set of points it is, in fact, valid for all z.
LEMMA 4.2. re(z) < + ~ if and only i / z is an ordinary point o / G or the fixed point of a parabolic element in G.
L E ~ M A 4.3 (The M i n i m u m Principle I). Let y be a closed Jordan curve in the finite complex plane and let D be the interior of y. Then
m i n re(z) >~ m i n re(z). (4.3)
zeD Ze ~'
Further, if V(No) c D and w E V(No), then
~*(V) = re(w) > rain re(z). (4.4)
ZE~'
Lv, M~A 4.4 (The M i n i m u m Principle l I ) . Let a and b be points in V(N~) N R 1 with a<b. I / x E ( a , b) then
re(x) >~ rain {re(a), re(b)} (4.5)
with equality holding if and only if x e V(N~).
LE~IMA 4.5. Suppose that V EG and zE V(N~). Then r e * ( V ) - I <~re(z) <~re*(V).
L E ~ M A 4.6. For n>~l the set A ~ = { z : re(z)>~n} is open. Further, r e = n - 1 on ~ A ~ - L . W e r e m a r k i m m e d i a t e l y t h a t t h e u p p e r b o u n d in L e m m a 4.5 follows i m m e d i a t e l y f r o m (4.2) a n d we shall need this before p r o v i n g L e m m a 4.5. Also, with reference t o L e m m a 4.6, it is false t h a t re = n - 1 on ~A~. This is easily seen as L e m m a 4.2 implies t h a t a n y parabolic v e r t e x p is a point of a c c u m u l a t i o n of points at which re= oo; t h u s re(p)< + oo a n d pE3A n for all n. I t is true, however, t h a t re = n - 1 on t h e b o u n d a r y of each c o m p o n e n t of A~ a n d this will be p r o v e d in t h e proof of L e m m a 4.8.
Proof o] Lemma 4.2. I f z is an o r d i n a r y point of G, t h e n z E V(N~) for some V in G.
T h e same holds if z is fixed b y a parabolic element in G [13, p. 149] a n d so in b o t h cases re(z) <re*(V) < + oo.
N o w suppose t h a t re(z) < + oo. The result is obvious if re(z) = 0 , t h u s we assume t h a t
242 A.F. BEA.RDON
re(z) ~> 1. T h e d e f i n i t i o n of z implies t h a t t h e r e exist sequences z~ a n d V~ w i t h Zn E V~(No) , Zn->Z a s n-->-c<) a n d
~r(z) = re(zn) = ~*(Vn). (4.6)
W e first rule o u t t h e p o s s i b i l i t y t h a t t h e r e are i n f i n i t e l y m a n y d i s t i n c t V~ i n t h e sequence.
I f so, we m a y consider a s u b s e q u e n c e a n d relabel; e q u i v a l e n t l y we a s s u m e t h a t V~ =4: Vm if n ~=m. As ~*(Vn) >~ 1, each V~ is a d j a c e n t to some U~ w i t h ~r*(U~) =z*(V~) - 1 a n d so t h e r e exists a sequence w~ satisfying
w~e v~(2vg) n v~(;v~).
As t h e V~ are d i s t i n c t a n d as co ENo, t h e e u c l i d e a n d i a m e t e r of V~(No) t e n d s to zero as n - + co a n d so w~-+z as n - + c~. This implies t h e existence of a sequence w~ w i t h w~E Un(No)
a n d Wn~Z as n---> ~ . ' T h u s
7~(z) ~< lim re(w') = z*(Vn) - 1
which c o n t r a d i c t s (4.6). T h u s t h e r e exists a V i n G w i t h Vn= V for i n f i n i t e l y m a n y n.
F o r these n we h a v e z~ E V(No) a n d zn-+z as n - + ~ . T h u s z E V(N~) a n d z is a n o r d i n a r y p o i n t or a fixed p o i n t of some p a r a b o l i c e l e m e n t i n G.
P r o o / o / L e m m a 4.3. W e first establish (4.4). Suppose t h a t w E V(No) a n d V(No) c D and, for convenience, p u t 7~*(V) = n a n d V = V~. I t follows t h a t V~ is a d j a c e n t to some V~_I w i t h ~ * ( V ~ - I ) = n - 1 . This process c a n be c o n t i n u e d a n d so we c o n s t r u c t a sequence of e l e m e n t s
I = Vo, V~ ... V ~ = V
i n G w i t h ~r*(Vr) = r a n d Vr a d j a c e n t to Vr+ 1 (r = 0 ... n - 1). This implies t h a t t h e r e exist p o i n t s w 0 .... , w,_ 1 such t h a t
w~eV~(;vg)n v~+l(;v~) ( r = 0 ... n - l )
and so K = [ 0 V~(No)] U {~o . . . w~_l)
r = 0
is arcwise connected. F u r t h e r , we h a v e re ~< n - 1 o n
n 1
K 1 - [ O v~(N0)] u (w0 . . . w~_~}
r = 0
as 7~(wr) ~<r. N o w c o n s t r u c t a simple arc ~ l y i n g i n K a n d j o i n i n g w (inside y) to co (out- side y). I t follows t h a t z meets 7 a t a p o i n t Zl, say, a n d b y o u r i n i t i a l a s s u m p t i o n z I E K 1. T h u s
re* (V) = n > ~(zi) >~ m i n ~(z)
z ~ ,
a n d this is (4.4).
I N E Q U A L I T I E S F O R C E R T A I N EUCIffSIA]K GROUI~S 243 To prove (4.3) define
and note t h a t if z E D, t h e n
D* = O n U V(No)
V ~ G
z(z) = lira inf ~r(w). (4.7)
w-->z, w 6 D *
L e t wED*, then wE V(No) for some V. If V(No)C D then b y (4.4),
~(w) = :r*( V) > rain Jr(z).
z ~ ,
The alternative hypothesis, namely V(No)~Z D, implies t h a t V(No)R V = O and so :r(w) = :r* (V) ~> min ~r(z).
Ze~'
I n a n y case, then, this last inequality holds and we see t h a t ~r >~ minz~r :r(z) on D* and hence, b y (4.7), on D. This establishes (4.3) and completes the proof of L e m m a 4.3.
Proo/ o/ Lemma 4.4. This follows easily from the Minimum Principle I. We join a to b b y a J o r d a n arc 71 which lies entirely in V(N0) f) {Ira (z) >0} except for the end-points a and b. Then Yl U ~)~ is a closed J o r d a n curve y lying entirely in V(N~). I f x 6 (a, b) then x lies inside y and so b y (4.3),
~(x) ~> min~(z).
ZE~
As ~<~r*(V) on V(N~) with equality on V(No) we see t h a t
rain ~(z) = rain {~(a), ~(b)} (4.8)
Zey
and so (4.5) follows. Suppose now t h a t equality holds in (4.5). We can find sequences zk and Vk such t h a t zkEVk(No), VkCG, ~(%)=r and z k ~ x as k - + ~ . I f Vk(No) c D, then (4.4) (with V = Vk and w =zk) and (4.5) (with equality holding) contradict 7r(zk) =~z(x). Thus Vk(N0) ~= D and so, as Vk(No) meets D, we see t h a t Vk(No) meets 7. This implies t h a t Vk(N0) meets V(No) and so Vk= V for all/c. Thus zkE V(No) and so xE V(N~) as required.
Proo/o/Lemma 4.5. To establish the lower bound we consider two cases.
Case 1. Suppose t h a t z ~L. T h e a there exists a finite m a x i m a l subset V1 ... Vs of G with, say, V = V1 and a neighbourhood N of z such t h a t
v (No), tJ (4.9)
r ~ l r = l
This implies t h a t each Vr is adjacent to VI( = V) and so b y (4.1) and (4.2),
244 A. ~. BEARDO~
7t*(Vr) ~>~*(V)-I
(1 <~r<~s)a n d ~(z) = m i n {~*(Vl) ... 7~*(Vs) }.
T h e second i n e q u a l i t y in this case is a trivial consequence of these last two results.
Case 2. Suppose t h a t z EL. The u p p e r b o u n d for z(z) established above together with L e m m a 4.2 shows t h a t z is t h e fixed p o i n t of some parabolic element P in G. As z EOV(N0), there exists a horocycle H (an o p e n disc lying in t h e u p p e r halLplane a n d h a v i n g z on its b o u n d a r y ) with
+ o 0 r ~ - o o
Next, choose a point x 1 in a free side of V(No) a n d note t h a t t h e circle y h a v i n g t h e segment with end-points x 1 a n d z as d i a m e t e r lies entirely in V(No) O {z}. F o r each integer r, write
~r = p r ( y ) a n d let D r be t h e interior of Yr- ~For sufficiently large r, s a y r = k, yr a n d y_r lie on different sides of z (which lies on R 1) a n d t h e set
Dk U D-k U ~,'e U y_,~ U [ U P ' V ( N ; ) ]
Irl<k
covers a n e i g h b o u r h o o d N of z. N o w let T(N0) intersect N. T h e n either T = PrV for some r satisfying ]r] ~</c or T(No) is c o n t a i n e d in either De or D k. I f T(No) ~Dk, we deduce f r o m (4.4) t h a t
~*(T) >~ 1 + m i n ~(w) = 1 + m i n {~(z), ~*(PeV)} f> 1 + rain {~(z), ~*(V) - 1}
w e ~ k
= m i n {1 + ~(z), ~* (V)}
t h e last inequality holding as for each r, V a n d p r V are a d j a c e n t (at z). A similar i n e q u a l i t y holds if T(N0) c D_ k a n d t h e same reason shows t h a t if T = p r V for some r satisfying [ r[ ~ k, t h e n
~* (T) >~ ~* (V) - 1.
T h u s in all cases ~*(T) >~ m i n {1 +~(z), z * ( V ) - 1 } a n d so
z(z) = lim inf 7~(w) >~ m i n {1 + ~(z), ~* (V) - 1} = z* (V) - 1
w.->z, WEE
as required.
(E = U T(No) n N)
TEG
Proo/o/Lemma 4.6. L e t zEAl, t h e n there exists a n e i g h b o u r h o o d N of z such t h a t z~>n on UvEGV(No)NN. I t follows f r o m (4.2) t h a t 7e~>n on N a n d so A~ is open.
I N E Q U A L I T I E S F O R C E R T A I N F U C H S I A N G R O U P S 245 N o w suppose t h a t z is a n o r d i n a r y p o i n t on 8An. T h e n t h e r e exists a m a x i m a l finite subset V1, ..., Vs of G satisfying (4.9) a n d b y (4.1) a n d (4.2),
~r(z) = r a i n {~r*(Vr): r = l ... s} > m a x {zr*(Vr): r = 1 .... , s } - I ~ > n - 1 as z~(An)'. As z~An, 7r(z)~<n-1 a n d so : ~ = n - 1 on 8 A ~ - L .
Our proof of T h e o r e m 2 depends u p o n a detailed e x a m i n a t i o n of t h e topological a n d m e t r i c a l properties of t h e sets An. Clearly t h e sequence of sets A~ is m o n o t o n i c w i t h n, t h u s we can label t h e c o m p o n e n t s of An as A(i 1 ... in) in such a w a y so t h a t
A(i 1 .... , i~)~ A(il, ..., in, i~+1).
F o r t h e sake of b r e v i t y we introduce t h e following notation. W e denote b y I~ (n>~l) t h e set of n-tuples i = ( i l ... i~) for which A(i 1 ... i~) is defined a n d rewrite A(i 1 .... , in) as A(i); we also write I for [J ~ 0 I ~ . :Next, we denote b y Z(i) t h e set of l - t u p l e s j = (~) for which A(i 1 ... i~, j) is defined a n d write A(i, j) for A(il, ..., i~, j) where jEZ(i).
I n order t o proceed w i t h t h e e x a m i n a t i o n of t h e sets A~ we m a k e t h e following defini- tions. W e write
A~(i) = {z EA(i): ~r(z)= n}
a n d n o t e t h a t A(i) is t h e disjoint union of t h e sets
A~(i), LI A(i, j) (4.10)
j e Z ( i )
N e x t , we write ffn=AnA R 1, a ( i ) = A ( i ) N R 1 a n d a,~(i)=An(i)N R 1.
Finally, we denote b y G(i) t h e set of V in G with the properties (i) V ( N o ) C A(i) a n d (if) ~r*(V)=n where iEIn. I f UEG, if
U(N0) [~ A(i) 4 ~D a n d if ~r*(U)=n, t h e n UCG(i).
T h e outline of t h e r e m a i n d e r of t h e proof of T h e o r e m 2 is as follows. W e see f r o m (4.10) t h a t a(i) consists of t h e disjoint union of t h e sets
a,~(i), IJ a(i, j). (4.11)
j e Z(i)
W e first p r o v e three l e m m a s which describe G(i) a n d t h e topological properties of t h e a(i).
A f t e r this, we need t w o m o r e l e m m a s which give i n f o r m a t i o n on t h e m e t r i c a l p r o p e r t i e s of t h e sets a(i). T h e n (4.11), these last two l e m m a s a n d L e m m a 4.1 yield the required result.
L E M M A 4.7. (i) The a(i) (i E In) are disjoint open intervals and hence are the components O] (Tn,
246 x . F . B E A I ~ D O N
(ii) An(i) is the closure relative to A(i) o/ (Jwa(i)V(No).
(iii) Each component o] an(i) is either (a) a parabolic vertex on DV(No) ]or some V in G(i) or (b) the closure relative to a(i) o] a finite union o/intervals Vr(Tr) where Vr E G(i) and the zr are/ree sides in N o.
LEZ~M~t 4.8. Each interval a(i, j) abuts (and so lies between) intervals V(~) and U(vl) where U and V are in G(i) and where each o / v and T1 is a/ree side in N o or a parabolic vertex on ~N o.
(i) I], /or some choices o/ U and V, we have U = V then this choice is unique and A(i, j) = V(A(r)) ]or some r in 11.
(ii) I / , / o r all choices o] U and V, we have U=h V, then U and V are unique and A(i, j) = V(Z) where ~ is a component o/the complement o / N ~ O (V-1U) (N'o). Further, this latter set contains at least one ordinary point.
T h e transformations U and V in L e m m a 4.8 are not necessarily unique. However, (i) and (ii) do show t h a t if there is more t h a n one possible pair (U, V), t h e n among all such possible pairs, there is a unique pair with U = V. L e m m a 4.8 also implies t h a t A(i, j) is b o u n d e d b y one (if U = V) or two (if U 4 = V) regions U(No) and V(No) and t h a t these regions are unique if chosen according to (i) and (ii). Typical situations in cases (i) and (ii) are illustrated in figures 1 and 2 below.
~ V(~)
~0,J) Fig. 1
V(No)
V(
a a ( i , j ) b
Fig. 2
INEQUALITIES FOR CERTAIN ~FUCItSIAN GROUPS 247 I n t h e course of t h e proof of L e m m a 4.8 we shall a p p l y the t r a n s o r m a t i o n V - i to the situation described in figures 1 a n d 2. This leads to the situations illustrated in figures 3 a n d 4 respectively. I n particular, these figures illustrate t h e A(r) a n d Z occurring in (i) a n d (ii) of L e m m a 4.8.
v ~(a)
2vo= v-~(V(No))
Fig. 3
V-~(b)
No = V ~(V(~'o)) V ~(w*)~ V ~(~)
Y-~(a) V ~(b)
Fig. 4
L ~ r ~ i 4.9. There exist8 a positive integer K (depending only on G) such that/or each i in I, G(i) in the disjoint union
G(i) = G~ t3 ... tJ G~(i) (4.12)
where G~ has at most K elements and where Gr(i) (1 ~r<~K) is a subset of a set o / t h e / o r m {TP~ Vr: n an integer} where T, Pr, Vr are all in G, Pr is parabolic and where P i ... PK, Vi ... V~: depend only on G.
P r o o / o f Lemma 4.7. We note t h a t a n ~ [.]ieLa(i), t h u s we need only show t h a t t h e a(i) are connected for t h e y are obviously disjoint relatively open subsets of R i. Suppose n o w t h a t a a n d b are two points in a(i) with a < b . T h e n as A(i) is open a n d connected we can join a to b b y a curve ~ in A(i) in such a w a y t h a t ~ consists of only finitely m a n y straight line segments. As V(No) is s y m m e t r i c with respect to t h e real axis for all V in G we have
~(z) =~(~) a n d so /k~ a n d t h e A(i) are also s y m m e t r i c with respect t o t h e real axis. I f we n o w let p be the reflection of y in the real axis we see t h a t e v e r y x in (a, b) either lies on 7 or lies inside a closed J o r d a n curve consisting of an arc of y and an arc of ~. I n t h e first
