THE SINGULAR POINTS OF THE FUNDAMENTAL DOMAINS FOR THE GROUPS OF BIANCHI
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Authors Woodruff, William Munger, 1936-
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/284851 This dissertation has been microfilmed exactly as received 67-11,360 WOODRUFF, William Munger, 1936- THE SINGULAR POINTS OF THE FUNDAMENTAL DOMAINS FOR THE GROUPS OF BIANCHI.
University of Arizona, Ph.D„ 1967 Mathematics
University Microfilms, Inc., Ann Arbor, Michigan THE SINGULAR POINTS OF THE FUNDAMENTAL
DOMAINS FOR THE GROUPS OF BIANCHI
by
William Munger Woodruff
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF MATHEMATICS
In Partial Fulfillment of the Requirements For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
1967 THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
I hereby recommend that this dissertation prepared under my
direction by William Munger Woodruff
entitled THE SINGULAR POINTS OF THE FUNDAMENTAL DOMAINS
FOR THE GROUPS OF BIANCHI.
be accepted as fulfilling the dissertation requirement of the
degree of Doctor of Philosophy.
Dissertation(director Date
After inspection of the dissertation, the following members
of the Final Examination Committee concur in its approval and
recommend its acceptance:*
/yi • S» Pay f-L* o*.
~~7~ S-rry7
*This approval and acceptance is contingent on the candidate's adequate performance and defense of this dissertation at the final oral examination. The inclusion of this sheet bound into the library copy of the dissertation is evidence of satisfactory performance at the final examination. STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable with out special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be grant ed by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, per mission must be obtained from the author.
SIGNEDoo ACKNOWLEDGEMENTS
The author wishes to express his gratitude to Dr. Harvey Cohn for suggesting the study of the groups of Bianchi and for giving guid ance during the entire work.
This paper was prepared under the partial sponsorship of
National Science Foundation grant G-7412.
iii TABLE OF CONTENTS
Page LIST OF ILLUSTRATIONS v
LIST OF SPECIAL SYMBOLS vi
ABSTRACT vii
Chapter I 1
1. Poincarfe's extension to H 2 2. An alternate derivation of the equations 5 3. Classification and fixed points 7 4. The isometric sphere 9
Chapter II 14
1. Discontinuous and discrete groups 14 2. Parabolic cusps 18
Chapter III 25
1. The fundamental domain 25 2. The boundary of the fundamental domain 30
Chapter IV 36
1. The singular points of the groups of Bianchi 36 2. The fundamental domain of r(u>) 43 3. Conclusion 60
REFERENCES 61
iv LIST OF ILLUSTRATIONS
Figure Page
1. The fundamental domain of rn(i) . ••> 45
2. The fundamental domain of T ( ) 45 •o i 3. The fundamental domain of T (OJ) where D • 2 or D > 3 46 00
4. The fundamental domain of T(i) 46
5. The fundamental domain of r( 1 + ) 47
6. The fundamental domain of r(i/F) 48
7. The fundamental domain of T(i/5) 49
8. The angular sector at an edge 53
9. The sector at a dihedral vertex 53
10. The sector at a tetrahedral vertex 54
11. The sector at an octahedral vertex 54
12. The neighborhood of 00 for r(i) 55
13. The neighborhood of 00 for r( ^ ). . . 55
14. The neighborhood of a singular vertex for r(u>) where D ^ 1 or 3 55
15. The initial identification in r(i) 58
16. The final identification in T(i). . . 58
17. The identifications in I*( * )
18. The fundamental domain of T(i/2) as a polyhedron 59
v LIST OF SPECIAL SYMBOLS
Page 1
5
5
9
12
13
14
14
18
19
19
19
19
19
27
27
27
36
36
38
38
vi ABSTRACT
Poincare (1883) gave a method for extending the action of
SL(2,C) to the half space £,ri,C with £ > 0. In this space, a group acts discontinuously if and only if it is discrete. A new proof of this is given in Chapter II. Of special interest are those discrete groups whose affine subgroups form a module with a two element base. These
are said to have • as a parabolic cusp. In Chapter III, the method of
isometric circles is extended to space to give isometric spheres and
the following theorem is proved. Theorem. Let T be a discrete group
for which « is a parabolic cusp. Then T has a fundamental domain
bounded by planes and isometric spheres. A discussion of the boundary of the fundamental region is given for groups with «® as a parabolic
cusp.
A group r(a>) of Bianchi is a group of unimodular matrices whose
coefficients are the algebraic integers of an imaginary quadratic field
K(OJ). The main result is the proof of a conjecture of Bianchi. Theorem.
The number of singular vertices of the fundamental domain of r(u>) is
equal to the class number h of K(u>). This result and the method of
isometric spheres are used to construct some of the fundamental domains
for r(w), and to prove the following topological properties of the funda
mental domains. Theorem. The fundamental domains are manifolds for
D » 1 and D »3*and for any other value of D, they are pseudomanifolds
which are manifolds at every point except the h singular vertices. The
vii viii
homology groups at the singular vertices of the non-manifolds are those
•f a torus. Theorem. Let h be the class number of K(u), and let x be
the Euler characteristic of the fundamental domain of r(<•>). Then
C 0 if D - 1 or 3, X - ] ( h if D - 2 or D > 3. Chapter I.
A group r •» {S,T,...} is said to act in a topological space
X, if there is a map
$(T): X -*• X such that
(i) (ii) (S)°(f>(T) «* <|»(ST) for every S and T in r. Usually, one omits S(T(x)) • (ST)(x). The group elements in r are called transformations of X. The group SL(2,C) - ad - be - 1, a,b,c,d complex acts in the Riemann sphere ( i.e. the plane with«) by setting The group SL(2,R) acts in the upper half plane in the same manner. In his memoire on Kleinian groups, Poincar£ (1883) gave a method whereby each T in SL(2,C) would define a homeomorphism of the closed upper half space £,n,C with 11 i ? 1 0 which would agree with the l action (1) in the £,n plane. That is, the action of SL(2,C) can be extended from the plane to half space. Let H denote the half space £,n,C with 00 > C > 0. H will denote the closed half space £,n,C with <>>2 t £ 0, and the set 1 2 H - H will be called the boundary of H. It is the boundary of H as a subspace of H and is the complex ( Riemann ) sphere. 1. Poincard's extension to H. The action of a transformation T in SL(2,C) can be extended from its definition (1) in the £,n plane to an action in H in the following way. T is considered as being the result of the successive inversions in a sequence of circles in the £,n plane ( four inversions will suffice, see Carath£odory (1954) §54 ). For each circle of in version, take the sphere in space which has the same center and the same radius as the circle under consideration and perform the identical sequence of inversions, only this time with respect to the spheres. The result is a homeoraorphism of H which agrees with the action of T in the £»rt plane. Since T can be represented in many ways as a succession of inversions in circles, it is necessary to show that the result in three dimensions does not depend upon the sequence of inversions used. Let T be a point in H. Through T we may pass three distinct spheres which have their centers in the £,n plane. They cut the £,n plane in three distinct circles Cj, C2 and C3. After the inversions are per formed, the circles Cj, C2 and C3 are changed into three other distinct circles cj, C2 and C3 in the £,n plane, and these circles depend only on T and not on the sequence of inversions. The three spheres with the | I V same centers and the same radii as , C2 and C3 intersect at a point T' - T(T) which is thus independent of the sequence of inversions used to derive the action of T. 3 Since inversion preserves angles and maps spheres ( and planes ) onto spheres ( and planes ), the action of T also preserves angles and maps spheres onto spheres. Since T maps H onto H, we may at times consider the action of T only in H instead of in H. It is clear that SL(2,C) does not act transitively in H, because a point of the boundary of H cannot be mapped to an interior point by a homeomorphism. It will be shown later, however, that SL(2,C) does act transitively in H ( see p. 13). Let us derive the analytic expression for the extension of the action of T to H. For real numbers A, B, C, D and for w complex, the equation Aww + B(w + w) + iC(w - w) + D » 0 represents the most general circle in the plane. If we set E • B + iC this circle becomes (2) Aww + Ew + Ew + D = 0. Now suppose that this circle has arisen as the image of some circle in the plane under the transformation T where w - T(z) - Sri' ®d - be " 1. The circle from which (2) arose is the circle A(az + b)(az + b) + E(az + b)(cz + d) + E(az + b)(cz + d) + D(cz + d)(cz + d) • 0. Expanding this gives A(aazz + abz + abz + bb) + E(aczz + adz + bcz + bd) (3) + E(aczz + bcz + adz + bd) + D(cczz + cdz + cdz + dd) « 0. The spheres in space with the same centers and the same radii as these circles can be found by replacing, zz by p'>2 _ K2 + n2 + ^2 and ww by p,2«C'2+n,2+C'2 where z • £ + in and w » + in'. The image of (£,n,C) will then be (S'jn'.t1). With these substitu tions, equations (2) and (3) become (4) Ap «2 + Ew + Ew + D 0, and aap 4- abz + abz + bb acp + adz 4- bcz + bd + E ccp2 + cdz + cdz + dd ccp2 + cdz + cdz + dd (5) g ~acp 2 + bcz + adz + bd + + D 0. ccp2 + cdz + cdz + dd Since E = B + iC and E • B — iC where B and C vary independently, E and E may be considered as separate ( real ) variables. But (4) and (5) are to represent the same sphere for all real values of A, B, C and D. Thus .'2 aap2 + abz + abz + bb aa£2 + (az + b)(az + b) ccp2 + cdz + cdz + dd cc£2 + (cz + d)(cz + d) acp2 + adz + bcz + bd ac£2 + (az + b)(cz + d) (6) < w m ccp2 + cdz + cdz + dd cc£2 + (cz + d)(cz + d) acp2 + bcz + adz + bd acC2 + (az + b)(cz + d) w ccp2 + cdz + cdz + dd ccC2 + (cz + d)(cz + d) These equations reduce to (1) if C « 0. By separating either of the last two equations into real and imaginary parts, one can solve for and n' in terms of C, n and Q. It is possible to find c'2 from c'2 = p'2 - ww. The result is 2 ~ t2 . o - zz cci;2 + (cz + d)(cz + d) But t2 - p2 - zz and both G and c' lie In the upper half space H, so we can take positive square roots to get (7) ccC2 + (cz + d)(cz + d) Often, (£,n,C) shall be written as (z,£) or as T, and thus (z', C') - T(z,C), T' - T(T), or T' - TT. 2. An alternate derivation of the equations. There is a non-geometrical way of arriving at the equations for the action of SL(2,C) in space. Let C2 be two dimensional complex space with the basis r a » e2 An element in C2 will be written as a column vector so that Vl + Z2£2 Let T be in SL(2, c) T : C 2 + C2 via W1 z a b - T , ad - be • 1. z c d iW2, The conjugate map T is represented by the matrix a b c d The two maps T and T define a tensor product map T®T : C2®C2 -*• C2®C2 - C\ As a basis for C1* ** C2® C2 we may take 1\ /0 /0\ fo\ 0 1 0 0 , e, ®E, - 0 » El*Pe2 " 0 • E2 e2 " 0 0 \o/ 1 Any element of C1* may be written as z e e + z e e + e ZjE^EJ + 2 i ® 2 3 2® i Zite2® 2* The matrices for T, T, E^Ej, E^E^ E2®Ej and E2®E2 give the following representation for the tensor product T®T. Wj\ aa ab ba bb \ z, \ w2 ac ad be bd w3 ca cb da db w,. cc cd dc dd Introduce non-homogeneous coordinates into C*4 by taking the quotient of each of the components by the last component. Let Wo w3 Z2 Z3 W1 " ^ • W2 ' W3-r" ' Z1 » Z2 " z, W.,4 w^ "4 "U ^ The mapping T®T then induces the following transformation in this three dimensional complex projective space. aaZ, + abZ2 + baZ3 + bb W, ccZ + cdZ2 + dcZ3 + dd acZ + adZ2 + bcZj + bd (8) W, ccZ + cdZ2 + dcZ3 + dd caZ + cbZ2 + daZ^ + db w, ccZ + cdZ2 + dcZ3 + dd These equations actually define an action of SL(2,C) in the complex 7 projective space because these equations define a homeomorphism, and because (T®T)(S®S) • (TS)<£>(TS). The equations in (8) are nearly the same as those in (6). If one computes Wj - W2W3, one finds that Z| Z2Zg (9) WJ - W2W3 = —- — . 2 (ccZj + cdZ2 + cdZ3 + dd) We are interested in the case where Z2 = Z3 °» z. In this case Wi is real if and only if is real, and - W2W3 is positive if 2 and only if Zl - Z2Z3 is positive ( from (9)), Now set Zj » p , 2 Wj «• p' and W2 = w to obtain the equations (6). 3. Classification and fixed points. It is standard to classify the 2*2 matrices by considering them as linear fractional transformations. One finds the fixed points in the plane, maps them onto 0 or 00 and arrives at the usual classifications: elliptic, parabolic, hyperbolic and loxodromic. This proceedure is the same as the proceedure for finding canonical forms for the 2x2 matrices under similarity. The standard facts are in Ford (1929) or Lehner (1964). A matrix T in SL(2,C) is non-loxodromic if and only if the trace a + d of T is real. In this case, T is elliptic if and only if |a +d| <2, parabolic if and only if |a + d| => 2, hyperbolic if and only if |a + d| > 2. One can use the same classification proceedure even though T acta in H, for suppose S puts T into canonical form. Then (S®S)(T®f)(S"Ie> S-1) - (STS-1)® (STS-1) \ 8 and it is possible to put T and T into canonical form simultaneously. Let us find the fixed points of T as it acts in H. Since T extends the action of SL(2,C) from the £,n plane, we may first find the fixed points in the plane and map them to 0 and ® ( or to » if there is just one ) by a transformation in SL(2,C). Suppose that this has been done and that in addition there is a fixed point (z',c') = T(z,c) • (z,i;) in H so 5 j* 0. If T fixes only «® in the 5»n plane, then z» . az + b m ^ Thus |d| =1, so z' = adz + bd and b • 0. d dd If T fixes 0 and 00 in the £,n plane, then a 0 0 d 22 2J_0 In either case, T is of this form. Now z' - ~ « Ke , where K > 0 and « -5- • 5. Again |d| « 1 and |a| ° 1 from ad • 1 making K - 1. dd The transformation T is i0 -ie 0 e and is elliptic, fixing the entire line from 0 to <*> perpendicular \o the S,n plane. This shows that the only transformations with fixed points in H are elliptic and that they fix a semicircular arc. The diameter of this circular arc is the line segment joining the fixed points of T in the £,n plane and the arc lies in a plane perpendicular to the £,n plane. 4. The isometric sphere. Let us find the Jacobian of the transformation T as given by (6) and (7). If c j1 0, I can be factored as a,1 b - o\ 0 -1' c fl ^c T - C V3T2T d 0 1 0 c 1 0 0 1; If c • 0, then d ^ 0 and /I li / b 0 1 bd ' d m d d , 0 d 0 i 1° 1 ( To find the Jacobian ^ • , which we shall write as we take the Jacobians of each of the transformations T^, T2, T3and T4 and multiply them together. The Jacobian of T1 is 1 because this transformation is a trans lation of upper half space. For T2 we have V + in' - T,(z) - —+ln + ln , c' - t (c) - 1 1 o o 2*• 1 S2 + n2 + G2 and thus 2£2 - p2 -2en -2CC 1 2Kr\ P2 - 2n2 -2nC 3(z,5) .12 2K -2nC P2 - 2C2 For T3 we have _ £ + in - JL V + in' T3(«) C' T3(C) cc 10 and thus 1 1_ 1_ _i_ 2 „2 ~2 :2 -i) 3 Cz'. C' > i 0 3(z,t) 2 c2 1 cc (cc) Now put 2 a 2 (Z1•Cj) Tx( z , x, ) where p* - + n + t,\ £ + (z + (z + 4), (z2»^2^ *^2 1 » ^ 1 ^ » (23,^3) Tj(z2 »£2)» (Zl4 »£|<) ^^(23,^3). The composition of these gives 3(z',C') 3(z',C') 9(Z3,53) A(z2,?2) AFZ^Q) 3(z,C) 3(Z3,C3) 3(Z2,^2) 3(Z1,C1) 3(Z,C) 1 1 , (10) 1* •—- *1 (cc)3 p. [ccC2 + (cz + d)(cz + d)] 3 — —3 We clearly get the result (dd) when c » 0. The rule for the Jacobian of the product of two transformations S and T gives nn 3(ST(z) .ST(C)) _ 3(ST(z).ST(Q) 3 Equation (10) shows that the expression IV, • is invariant under the action of any transformation in SL(2,C). However in fact, it is easy to show that /(dO* + (dn)* + (del7 ds ds. = N t, Z is invariant under SL(2,C) and thus derive the value of the Jacobian. Let T and T + dx be in H. If, in a sequence of inversions, one performs an inversion with respect to a sphere with center (z,0), then pass a plane through (z,0), T and x + dr. We may assume that z • 0. If Tj is the inverse point to T with respect to the given sphere, it follows that dTl dx =* TI T But T1 ?1 1 - ai T so dTi dT C Now perform the sequence of inversions taking (£,n»C) to (C* »n' Clearly, dil dul da dll 5' c C' e Squaring and adding gives (dE')2 + (dn')2 + (dE*)2 _ (d£)2 + (dn)2 + (dt)2 or ds' ds C* " 5 The expression ds ds N is called the non-Euclidean metric in H, and it is invariant under any transformation of SL(2,C). Invariant ( non-Euclidean ) area and volume are defined by J A dA J J,, dV dAN " I7 N " T7 • This is the usual non-Euclidean metric in the upper half space H, so we may use the concepts of non-Euclidean geometry in H. In particu lar, non-Euclidean planes are planes or spheres perpendicular to the £,n plane. The points of the boundary of H are at an infinite non- Euclidean distance from any point of H and they are called singular points. The usual metric ds is unchanged by T if and only if £ = £r, and so from (7), ds is unchanged by T if and only if (12) ccC2 + (cz + d)(cz + d) = 1 Definition. That part of the sphere (12) which lies in H will be called the isometric sphere of the transformation T, provided c j4 0. If c - 0, then T will not have an isometric sphere. The isometric sphere of T will be denoted by I(T). Elements of arc which lie on this sphere will not change their length under T. If c » 0, then Euclidean are length is preserved everywhere. The isometric ball of T is given by ccC2 + (cz + d)(cz + d) i 1 , (z,C) in H and will be denoted by B(T). The intersection of the isometric sphere with the £,n plane is the usual isometric circle as given by Ford (1929). Finally, it should be noted that SL(2,C) acts transitively in H. That is, if (z,5) and (z',5') are two points in H, then there is a T in SL(2,C) such that (z',c') - T(z,q). It is sufficient to show that every point can be mapped to (0,1). By a translation, (z,c) goes into (0,O. The transformation o -/r\ X 0 / will then map (0,0 into (0,1). Chapter II. 1. Discontinuous and discrete groups. Let T be a group which acts in a topological space X. The group T is said to be discontinuous at a point- x in X if there is no y in X and no sequence T^ of distinct transformations of T such that T y converges to x. The group F is discontinuous in X if it is dis continuous at every point of X. Suppose that T is a topological group. Then T is called a discrete group if the topology on T is the discrete topology. In particular, we are interested in subgroups of SL(2,C) which act in H. SL(2,C) is a topological group under the usual topology which arises by considering the matrix (c d) ' 3d " bC * 1 as a point (a,b,c,d) of C4. However, the two matrices T and -T give the same transformation of II, s~ in order to topologize the group SL(2,C) considered as a group of transformations of H, we must identify T and -T. The topology on the transformation group is a metric topo logy which can be given by I IT|| = /|a|* + |b|" + |c|* + |d|* and d(T,S) = Max(]|T - S||,||T + S||). We shall use this latter topology on SL(2,C). Thus T is discontinuous if it is discontinuous in H, and T is discrete if the topology induced in T as a subspace of SL(2,C) is the discrete topology. To say that 14 r is discrete is equivalent to saying that no sequence Tr from T con sisting of distinct matrices converges to 1 0\ T 1-1 0 0 1 or 0 -1 * / ' The following two lemmas are immediate consequences of the discreteness of T in SL(2,C). Lemma 1. If T C SL(2,C) is discrete, then it is countable. Proof. Any uncountable set in a second countable topological space has an accumulation point. Thus if T is not countable, it is not discrete. Q.G.D. Lemma 2. Let T be a discrete group. There is a point T in H which is not a fixed point of any T in T. Proof. The fixed points of T are arcs of circles or are empty sets and these are both nowhere dense in H. There are only countably many such sets, so the Baire Category Theorem assures the existence of a non- fixed point T. Q.E.D. For a general group T acting in a topological space X, dis creteness and discontinuity are not equivalent. The standard example is Picard's group acting in the plane ( see Lehner (1964) p.96). However, for subgroups of SL(2,C) acting in H, we have the following fundamental theorem ( see Poincar£ (1883), Fricke-Klein (1897) p.98, or Fatou (1930) p.86). Theorem 1. A group T C SL(2,C) is discontinuous in H if and only if it is discrete. Proof. Let us first show that discontinuity implies discreteness. Assume that T is not discrete. Let T be a sequence of distinct 16 transformations converging to I or -I. Since we may assume that both T and -T belong to T, we may also assume that T -»• I, Let T be in n n e } n H. We have T T • T -+• T. It is possible that j for all n, but n n 1 n in any case, T is not discontinuous. To show that discreteness implies discontinuity in H, we assume that T is not discontinuous in H. Let T be a point of H at which T is not discontinuous. There must exist tq in H and a sequence of dis tinct transformations T of T such that T t t. By the transitivity n no' of SL(2,C) in H, there is an S in SL(2,C) such that tq - S(0,1). Set (z,c) • S 1(T) where C > 0 because T is in H. Now 1 S' Tn S(0,1) (z,C). We shall show that the transformed group S 1 T S is not discrete, and thereby conclude that f is not discrete. Set . a b S - S T S - " " , det S = 1. n n " 1 cn dn The S are all distinct and S (0,1) -»• (z,c). By equations (6) and (7) n n of Chapter I, + ? , c c + d d n n n n a c + b d n n n n — —- z , c c + d d n n n n and ad -be - 1. n n n n Both the sequences c and d must be bounded ( C > 0 ) and hence there n n exists a subsequence S cf S such that the c converge to some ni n ni 17 complex number c. There is then a subsequence of such that the dn 's converge to some complex number d. Relable this second subse- sequence as S . Thus the S are distinct and both c •* c and d •+ d. n n n n n Now ac +bd «• z(lcl2+|d|2)+e where e ->-0, nn nn 1 n' 1 n1 n n and c c + d d - _ ~—- where e' 0. nnnnc+ e n n Multiply these equations by cr and aR respectively and subtract the first from the second to get a (ad - b c )d •» ———rIL_ -_ c„ z(wl„ c 122 +u. \dA 1 z) + c e nn nnn c +e n'n1 n nn n a or d » —-—R - c z( c + d p) + c E n C"+ e' n'n' 1 n1 nn n As n + », 2 2 an [ d + cz(|c| +| |d ) ] C . Thus the a's tend to a finite limit a -*• a. Finally, since a , c and n J n n dn approach limits and a d - 1 ^ z(|c I2 + Id I2) + e - a c nn , v,n' 1 n1 n nn b - or b - , n cn n jd n we see that b tends to the finite limit b ( not both c and d •+• 0 ). n n n The sequence is such that » s* - a ; with det S* => 1. s n led 1 s is in Now S* need not be in the group S T S but n+j this group and is not I ( because the S are distinct ), and n S~* S + S*"1 S* - I. n+1 n Thus S 1 T S is not discrete and hence neither is the group r. Q.E.D. It is conceivable that T might not be discontinuous in H, but that it might be discontinuous in a subset of H. This is not the case Let T . T and T be as before. If T X = x for all n, then n' o no T 1 T, T *• x and T possesses infinitessimal rotations and is discon n 1 o o tinuous everywhere in H. Thus we may assume that TnxQ t x for all n. From the proof of Theorem 1, T T in SL(2,C). Let T* be in H, and let T* - T-1T* then T x* - T T-1T* x*. o n o n Corollary. If T is not discontinuous at a point of H, it is not dis continuous everywhere in H. Finally, let us note that if T is discontinuous in the plane then it is discrete and hence it is discontinuous in H. This puts a large number of discrete groups at our disposal for examples. 2. Parabolic cusps. Let T be a discrete group. Definition. The subgroup of T which consists of all the transfor mations T in T such that a b " X b 0 d 0 X_1 will be called the affine subgroup of T. The affine subgroup of T is the subgroup of T which fixes the point «®. The number X2 is the multiplier of T ( see Lehner (1964) p.70 ). The subgroup £ of consisting of all T in for which X2 - 1 will be called the translation subgroup of T. It is clear that the collection of multipliers of T forms an abelian group. 19 We shall write 1 0 _a t and RX ,-1 0 " L so that T - TXb R^. If the translation S « Ta R^ , Xj • 1 is transformed by T, the result is 1 Xb a Xb T* ST - Rx_! T~ T Rx T Rx - Rx-> T* \ Rx " V* T" Rx \ - t^r, . a1 If S is a translation given by a, then T 1 S T is a translation given —2 by X a. Since the multipliers form a group, there is a transformation in r with multiplier X -2, so if a gives a translation so also does X^a.o The translation group is a module over the integers Z and hence the numbers a which define T in F form a Z-module. This module is 0, oo has one generator a1 or has two independent generators aj and a2. Let [0], [aj] and [aj,a2] denote these modules. Definition. We shall say that 00 is a parabolic cusp of T if the translation module has two independent generators. If it has one generator, then 00 will be called a semi-parabolic cusp, and if the module consists of 0 alone, then « will be called a non-parabolic cusp. Let ® be a parabolic cusp. If X2 is a multiplier of r^, then 2 a±X € [aj,a2] , i - 1,2. Thus (1) a.X2 - f A..(X 2)a. where A..(X2) € Z. i j-1 ij 3 i-3 The map X2 -> (X2)- is an isomorphism of the group of multipliers into SL(2,C) so A^(X2) is a unimodular matrix. We may write (1) as 0 ml 12 2 X ^21 ^22 From the linear independence of the a^, we get 2 2 Thus X1* - (A^ + A22H2 + 1 = 0, and X2 is an algebraic unit of degree two. We must have |x| =1 because if not, then Tn ( or T n ) 0r> 9 n would have as a multiplier X ( or X" ), and the translations given 0_» O rl by a^X ( or ajX~ ) would be arbitrarily small. The only alge braic units of degree two which have modulus 1 are t 1, i i and t 1 ± i/3. ^ " Thus, when 00 is a parabolic cusp, the only possible multi pliers of r are + + * 1 * i * 1 t i/3 ± /? ± i - - i» » 2 ' 2 The transformations in Tm are either parabolic or elliptic. If » is a semi-parabolic cusp, then X2a£[a] and X2a - A(X2)a. Since X2 A(X2) is an isomorphism into Z, we see that A(X2) - ± 1, so X2 » ± 1. The transformations in are again parabolic or elliptic. If 00 is a non-parabolic cusp, then the transformations of 1*^ are of the form ' X 0 0 X'1 Discrete groups for which » is a non-parabolic cusp exist. As an example, the group generated by 21 , n « 0, t 1, ± 2, -2n 0 is discrete, and has as Its affine subgroup the transformations 2n 0 B , n - 0, ± 1, ± 2 ,-n n 0 2 These transformations are all hyperbolic. We shall be interested only in groups where « is a parabolic cusp. For these groups we have the following proposition: Proposition 1. Let » be a parabolic cusp of a discrete group T. Let T be given by t - (: 5 Then either c • 0, or else |c| > A > 0, where A depends only on the group T and not on T. Proof. Suppose to the contrary, that a b n n c n is a sequence from T with c / 0 and lim |c| = 0. We may assume that n n r* ft n n cr -• 0. Multiplying Tr by translations T and T gives a* b* n n an n x* = T T T^ n n c* d* n n a + a c b + 8 a + a B c + ad n n n n nn nnn nn d + 0 c n n n We can choose a and 3 so that the numbers n n 22 lie in the fundamental period parallelogram for the translation module of T. The norms of these numbers are bounded by the diameter 6 of this parallelogram, so that a - 1 n |a* - 1| + cl £ *|c n d - 1 n and d* + s. |c | < 6|c |. ' n ll 1 n1 - 1 n' Now Ib*c*| la* d* - ll 1 n n1 1 n n 1 |(a* - 1)(d* - 1) + (a* - 1) + (d* - 1)| 62 'CJ2 + 26'cn' * Thus b* «|c |(sjc^t + 2) , 1 n1 n n' so the sequence b* is bounded. The sequence b* must have a convergent subsequence which we shall assume to be b*, and we shall consider that T* has been chosen to give this subsequence. Set a* - 1 + where e ->0. Then n nk nk T*"1 T* - n k "nk nkj with -c*(l + e. ) + c.*(l + e ). "nk n k k n This converges to the identity for kin and n -*• <*>. Choose nQ such that |en| < ~ for n i nQ. Then > A-| C *| - — I c*L. 1 nk1 2 2 1 n1 2 1 k1 Now choose k(n) « k so that lc*l > 3 |c*| and thus obtain a sequence of distinct transformations which converge to I, none of which is I itself. Q.E.D. 23 Definition. A point t = (z»0) will be called a parabolic cusp of T if there is some S in SL(2,C) with S» » T such that «° is a parabolic cusp for S 1 T S, Similar definitions can be given for semi-parabolic and non- parabolic cusp points. Corollary. If T is a parabolic cusp of T and S is the transformation mapping •» -*• t, then for T in S *r with third entry c, either c = 0, or else |c| £ A > 0 where A depends only on T and S Proposition 2. Let T be a discrete group for which 00 is a parabolic cusp. Then any sequence of distinct isometric spheres whose centers lie in a bounded region of the 5,1 plane must have radii which tend to 0. Proof. Suppose that tr is a sequence from t with distinct isometric spheres whose centers lie in a bounded region of the 5,n plane. By Proposition 1, there is a A > 0 such that |c^| £ A >0. The radii 1 1 r 00 of the corresponding spheres is r = i r so rn £ — making n ' n' impossible. If r^ does not tend to zero, then there is a subsequence of T for which r •+• r > 0 and thus c •+• c t •». Call this subsequence n n n d T again. Since — lies in a bounded region, there is a subse- n dn d quence of T ( call it T again ) for which dfl d / ® and - »• - —. n Now consider ad.,—be., a.,b -ab.' n n+1 n n+1 n+1 n n n+1 Tn T"i,n+1 c d ., - c . d a.-id - cb._ n n+1 n+1n n n+1 n n n+1 where c d - c ,-d -»• 0. The modulus of the third entry is either n n+1 n+1 n J 0 or is £ A, so there is an N such that c d - c ,,d =0 for all ' n n+1 n+1 n n £ N. This shows that all the isometric spheres in the sequence Tr have the same center when n £ N. It also shows that for n Z N, the product T T is in T . Thus there are transformations v n n+1j. « s'n' s'n+1' , in T such that 00 . i ST - T ,. » S T . i • T _ n n n+1 n+1 n+1 n+2 Thus for any n N, there is an S in T with T = S T„. If J n 00 n n N a b n n S with la I «= 1. n i n' 0 a"1 n then an^ a.. + Sn c..N a n bXTN + 0n dNM ( -1 -1 , n N n N Since |<*n| • 1, all the isometric spheres of the Tr have the same center and the same radius when n £ N and this is contrary to assumption. Q.G.D. Corollary. If « is a parabolic cusp of T and if X is in H, then T is interior to, or on only a finite number of isometric spheres. Chapter III. 1. The fundamental domain. If T is any group of one to one maps of a topological space X onto itself, then one may define an equivalence relation in X by x~y if and only if there is a T in T with Tx «* y. A fundamental set in X is a complete set of representatives for X/~. For the cases in which we are interested, a fundamental set cannot be open. Instead of a fundamental set, we shall use an open fundamental domain. A fun damental domain of T relative to X is defined to be a non-empty open subset F of X containing no distinct equivalent points and such that every neighborhood of a boundary point of F contains a point of - F equivalent to a point of F. This is the same as requiring that F be a maximal open set which does not contain equivalent points. If F and f' are two fundamental domains with FCf', then F • f'. A fundamental domain need not be connected. By a Zorn's Lemma argument, one can easily show that a dis continuous group possesses a fundamental domain in H. Let J be the set of all open subsets of H which do not contain two equivalent points. Then J& is not empty and is partially ordered by inclusion. The union of any totally ordered collection from & is again in % and hence & possesses a maximal element which must be a fundamental domain. 25 26 The difficulty with this construction is that one has no control over the shape of the boundary of the fundamental domain. Furthermore, the fundamental domain might not be connected. We want to show that any discrete group T acting in H has a connected funda mental domain bounded by planes and segments of spheres having centers on the plane. There are two standard ways to do this for discon tinuous groups acting in the plane. One way is to use the hyperbolic metric and the other way is to use isometric circles. These two methods can be used in H. The easiest way in theory is via the hyperbolic metric in H. Let T be a point of H which is not a fixed point of T ( by Lemma 2 of Chapter II ). By the discontinuity of T there is a neighborhood of T which does not contain any two equivalent points. Since T is countable, the collection of points equivalent to T can be put into a sequence 7^,12 Let denote the non-Euclidean plane of points which are at equal non-Euclidean distance between T and T . Each P divides M n n H into two half spaces each of which is connected and non-Euclidean convex. The intersection F of all such half spaces which contain T is connected, non-Euclidean convex and is a fundamental domain. To see that F is a fundamental domain, suppose that Oj and o2 are closest to T and suppose that Ta1 « A2, then T must fix T which is not allowed. Thus 'fto two points of F are equivalent. On the other hand, every point of H is either equivalent to a point of F or to a boundary point of F. Clearly, F is open and non-empty and the boundary of F consists of the required type of plane and spherical segments ( possibly infinite in number ). Although the above method is simple in theory, it is difficult to use in order to obtain the equations of the bounding planes and spheres. In fact, these equations depend upon the point T chosen. It is convenient to have a method which gives the equations of a funda mental domain in terms of inequalities which are easily derived from the transformations of T. This can be done by the method of isometric spheres which is analogous to the method of isometric circles used in the plane. Definition. If T is in SL(2,C), the expression (1) 6(T,t) = [ccC2 + (cz + d)(cz + d)] 3 will be called the deformation of T at T. The deformation is just the Jacobian of the transformation T. From equation (11) in Chapter I, we have (2) <5(ST,T) - 6(S,TT) 6(T,T). The special case S «• T 1 gives (3) 6(T_1,Tt) - 6-1(T,x). Equations (1) and (3) have the following as an immediate con sequence. Lemma 1. If T has an isometric sphere I(T), then 6(|T,T) 1 according as T is inside, on, or outside of I(T). If T is outside of I(T), then TT is inside of I(T-1). Let Fj be a fundamental domain for This is always easy to determine when » is a parabolic cusp. Let k - U(b(t) | ter - rn > and let F2 • exterior of K in H. 28 Lemma 2. Each transformation of T maps K onto itself. • I »lll •ll»l-« • 00 Proof. Let a b 0 a"1 be in T , a ^ 0. Then it is easy to show that if S is in T - T , 00 00 T S T 1 is again in T - T . Let T be in B(S) so 6(S,T) i 1. Then CO 6(T S T^.TT) - 6(T,ST) 6(S T_1 ,TT) - |A|6 6(S,X) 6(T"1 ,TT) I |A|6 6-1(T,T) - 1. Thus T maps B(S) onto B(T S T_1). Q.E.D. We are now ready for the main theorem. Theorem 1. Let T be a discrete group for which ® is a parabolic cusp. Then F • F2 is a fundamental domain for T. Proof. F2 is open and non-empty ( because it contains a neighborhood of <*> ). Let t be in F0 and let T be in T - T . Since t is outside of ' Z oo each B(T), TT lies inside of B(T *) and thus outside of F2, showing that no two distinct points of F2 are equivalent under T - A fundamental domain for F1 is a prism with faces perpendicular to the plane erected over the plane fundamental domain for Thus F is open and non-empty. No two points of F^ are equivalent under - I, so no two points of F are equivalent under T. Now let T be in H ( so C > 0 ). We shall show that there is an element of T which maps T into F. First let us show that there is an element of T - which maps T into F2. If t is not in F2, it can lie 29 interior to certain isometric spheres in K, or it can lie on the boundary of certain isometric spheres of K. In either case, it lies interior to ( or on ) only a finite number by the Corollary to Propo sition 2 in Chapter II. Thus there are finitely many deformations at T that are greater than or equal to 1: (A) 6. - 6 =•...» 6 > { ,, i ... i 5 i 1, 12 s s-HL n where 1 < s < n and 6^ = 6(TJ,T). = 00 cannot occur and if all 6's are equal to 1, then T is in F. Suppose that 61 > 62 £ ... . By Lemma 1 and equations (2) and (3), we have {(T.TjT) - 6(T TJ.T) - 6(TTLFT) 6"1 < 1, so TjT is outside of I(T) for any T, and hence Tjt is outside of all isometric spheres. Since T^t is in H, its neighborhoods cannot meet infinitely many isometric balls. Thus there is a neighborhood of T^t excluding K - K, and hence Tjt is in F2« Now assume that in (4), s 2, 2. The loci 6^ = 6^, i ^ j, i,j = L,...,s are spheres or planes. We can move T a distance less than any given e > 0 to a point that avoids all the loci and is such that 6 is still less than 1 for any deformation other than and such that 6(T, ,x) > 6(T ,t) where 1 £ h £ s and s+1 % m S n. n m The 6(T, ,T) are now distinct for 1 i h s s and hence there is a k n with 1 £ k s s for which 6(Tk,x) is largest. The integer k is inde pendent of e because the loci 6^ = 6^ partition the neighborhood of T into disjoint regions and k is the same for all in a given region. By what we have already proved, ^2* Every e-neighborhood of T, T contains a T. r and hence meets F,. Thus T. t is in F?. k k o 1 k z Now let us show that there is an element of which maps any T in F2 into F. We may suppose that x is in F2. There is a T in such that Tx is in Fj. By Lemma 2, T maps K onto itself and hence it also maps F2 onto itself. Thus TT is in FjOF2 «• F, so every point in H is equivalent to a point in F. Finally, we must show that every point of the boundary of F has points of - F equivalent to points of F. If t is on the boundary of F and does not lie on an isometric sphere, this is clear because of the nature of and F^. If T lies on an isometric sphere, then points in a neighborhood of x lie interior to an isometric sphere and these are equivalent to points of F. Q.E.D. 2. The boundary of the fundamental domain. We shall assume that 00 is a parabolic cusp for T. Let us consider the boundary points of the fundamental domain F as determined by isometric spheres and planes which lie in H. By the Corollary to Proposition 2 in Chapter II, a boundary point x can lie on only a finite number of isometric spheres or plane faces of F^ We distin guish three cases: (i) x lies on only one isometric sphere or plane, (ii) T lies on exactly two isometric spheres or planes, (iii) T lies on three or more isometric spheres or planes. The boundary of F lying in H consists of at most a countable number of segments of isometric spheres or planes. The interiors of these seg ments ( taken as a subset of the isometric sphere or plane ) are called ( ordinary ) faces and consist of points in category (i). The points of category (ii) which lie on the intersection of the closure of two faces will be called an ( ordinary ) edge. The edges are open line or circular segments. A point of intersection of the closure of three or more faces will be called an ( ordinary ) vertex. These are the points of category (iii). As in the case of groups acting in the plane, there is the special case where a face is a segment of a sphere aris ing from an elliptic transformation T with T2 = I. The fixed points of T will be called an edge and they separate that part of the isomet ric sphere of T which bounds F into two congruent faces. The theory in H is similar to that for groups acting in the plane. Proposition 1. That part of the boundary of F lying in H is the union of countably many conjugate faces (f^,f^) such that (1) each face is a segment of an isometric sphere ( or the whole hemisphere ) or is a segment of a plane face of r^, i (2) to each face f^ there is a conjugate face f^ and a T^ in T with T.f. » f!. If f. and f. lie on different isometric i i i i J spheres ( or planes ), then T^ ^ T^, (3) f^ is not T-equivalent to a face other than f^, (4) f. and f' have the same Euclidean area. l i Proof. The proof is similar to the proof of Theorem 3B p.121 in Lehner (1964). It is clear that the following holds for ordinary edges. Proposition 2. F has at most a countable number of ordinary edges. F is bounded at an edge e in H by two isometric spheres, by an isometric 32 sphere and a plane of the fundamental domain for r , or by two such planes, and these planes or spheres meet in the closure of e. An image of e which lies in F is also an edge of F. The neighborhood of any point interior to e is made up of a finite number of images of F each of which has e as an edge. A similar proposition holds for ordinary vertices. Proposition 3. F has at most a countable number of ordinary vertices. F is bounded at a vertex v in H by a finite number of isometric spheres or planes which meet at v. An image of v which lies in F is also a vertex of F. The neighborhood of an ordinary vertex is composed of a finite number of images of F each of which have v as a vertex. There will be points of the Riemann sphere which are limit points of F. They belong to the boundary of F as computed in H. These singular points of a fundamental domain F in H can be regions which are as complicated as a plane fundamental domain, because a plane funda mental domain bounded by isometric circles extends to a fundamental domain in H bounded by spheres. Thus several categories of singular points may be defined ( see Lehner (1964) p.120 ) and F may possibly have singular faces, edges and vertices. Definition. The relation of equivalence under T partitions the ordin ary edges of F into equivalence classes called ( ordinary ) edge cycles, and it partitions the ordinary vertices of F into equivalence classes called ( ordinary ) vertex cycles. Proposition A. An edge cycle contains only a finite number of edges and a vertex cycle contains only a finite number of vertices. 33 Proof. The proof is similar to the proof of Theorem 4 A p.125 in Lehner (1964). Definition. If no edge in an edge cycle is fixed by an elliptic transformation, then the cycle will be called an accidental edge cycle. If one ( and hence every ) edge in an edge cycle is fixed by an elliptic transformation, then the cycle will be called an elliptic edge cycle. In the neighborhood of a fixed point T, the transformations fixing t are ( proper ) rotations. Thus the subgroup r of T fixing an edge e is a finite rotation group in a neighborhood of e, where all the rotations have the same axis. Thus Tmust by a cyclic group. Definition. The order of the cyclic group which fixes an edge e of an elliptic edge cycle is called the order of the elliptic edge cycle. The order of an edge cycle does not depend upon the edge e chosen. Proposition 5. The sum of the dihedral angles at the edges of an edge cycle is ~ if and only if the cycle is elliptic of order n ^ 1, and the sum is 2n if and only if the edge cycle is accidental. Proof. The proof is similar to the proof of Theorem 4C p.126 of Lehner (1964). Definition. If no vertex in a vertex cycle is the fixed point of an elliptic transformation, then the vertex cycle will be called an accidental cycle. If one ( and hence every ) vertex is the fixed point of some elliptic transformation, then the cycle will be called an elliptic vertex cycle. In the case of an elliptic edge cycle, the group fixing the edge was cycic because the only finite groups of ( proper ) rotations all having the same axis are the cyclic groups. At an ordinary vertex v, let rv denote the subgroup of T fixing v. Then consists of elliptic transformations which are rotations in a neighborhood of v. They are proper rotations because the Jacobian is always positive, and there must be only a finite number of them because T is discontinuous. The only finite proper rotation groups are the following: (i) the cyclic groups ^ of order n, (ii) the dihedral groups lDn of order 2n, (iii) the tetrahedral group IT of order 12, (iv) the octahedral group © of order 24, (v) the icosahedral group 3C of order 60. It is thus possible to classify the elliptic cycles by the group T which fixes one of the vertices in the cycle. Definition. An elliptic cycle is said to be cyclic of order n, dihedral of order 2n. tetrahedral, octahedral or icosahedral, if the subgroup of T which fixes any one of the verteces in the cycle is the like named group. This classification does not depend upon the point of the cycle chosen. By considering the manner in which the fundamental domains surrounding a vertex partition the neighborhood of the vertex, we have the following proposition. Proposition 6. The total solid angle subtended by the vertices of a vertex cycle is given by 35 1. 4tt if and only if the cycle is accidental, 4tt 2. — if the cycle is cyclic of order n, 2tt 3. — if the cycle is dihedral of order 2n, 4. if the cycle is tetrahedral, 5. -g- if the cycle is octahedral, 6. • if the cycle is icosahedral. Proposition 7. Let T be discontinuous, let the ordinary faces of F be denoted by f , f', n=l,2,..., and let T be such that T f • f . Then J n' n' ' ' n n n n T is generated by the set {Tn| n=l,2,...} . Proof. The proof is similar to the proof for groups in the plane. See Theorem 5D p.135 in Lehner (1964). Corollary. If F has a finite number of faces, then T is finitely generated. Chapter IV. 1. The singular points of the groups of Bianchi. Let K(/-D) be an imaginary quadratic field with discriminant - D. Let 1 + i> j 2 ^ if D s 3 mod 4, ^ i^D if D $ 3 mod 4. The integers of K(/-D) «• K(w) are of the form m + nw where m and n are rational integers. Let D if D = 3 mod 4, 4D if D jf 3 mod 4. Then J i/D = i/d if D = 3 mod 4, 0) - 0) ° \ I 2i/5 => i/d if D jS 3 mod 4. , , ., d + i»^d In both cases, u> • ^ • Let r(u>) be the group of all unimodular transformations which nave the algebraic integers in K(u) as coefficients. Definition. r(u) will be called a group of Bianchi. The group T(i) is called the Picard group. The algebraic integers in K(u>) form a discrete plane lattice so the group T(u>) is a discrete group and thus has a fundamental domain F(w). The affine group ^(w) has a two element basis [l,w] so « is a parabolic cusp and the fundamental domain F(u>) may be determined by isometric spheres. Bianchi has done this by other methods for a large 36 37 number of cases ( see Bianchi (1952) p.233 ). In each case, Bianchi noticed that the number of singular vertices of the fundamental domain was equal to the number of ideal classes of K(ui). He conjectured that this was true in all cases ( see Bianchi (1952) pp.181, 271 ). We shall prove this in general. Theorem 1. The number of singular vertices of F(u>) is equal to the number of ideal classes of K(w). To prove this we need several lemmas. Lemma 1. The only parabolic fixed points of r(u>) occur at «® and at (k,0) where k is in K(w). Every such point is a parabolic cusp. Proof. The parabolic fixed points of T lie in the 5,n plane. To be a fixed point of T, T must satisfy z » a ~ + ^ ^ = o or else C = °°. For parabolic transformations (a + d)2 - 4 = 0, so the parabolic transformations have at most the fixed points z » —7,d ~ 3 ,£•=() or else ? = °°. ic Thus the parabolic fixed points are 00 or are in K(w). On the other hand, let — be in K(w). Let a, 3, a be integers in K(u>) such that (x 8. a a Y 6 Then 1 + y6b y2b S_1 Tb S -y2b 1 - y6bj This transformation is in r(u>) and it fixes 1 - y«5b - 1 - y6b 6. Z " -2y*b " y ' Clearly the translation module of ^as a two element base, so 6 Y — is a parabolic cusp. Q.E.D. We now follow the method of Maass (1940). Let h be the number of ideal classes in K(u>), and let ai = i=l,...,h, be a set of representatives for these classes where we may assume that y^ = 0 and 6. *1. Let a. and $. be chosen from K(ui) so that 1 ii 'a. 1 1 det T^ = 1 , (i=l,...,h) with T1 = I. Ti yi Note that ou and 0^ are integers in K(u) only for i «• 1. Let U T. r(w). i-1 1 Lemma 2. If y and <5 are integers of K(u>) with (Y,<5) ^ (0,0), then there is a transformation T in L such that o a T—o • (Yo »<5 o) , Yo oy , 6o =06 ,» |o|i i < q where the constant q > 0 depends only on K(u>) , and the symbol denotes the second row of T . o Proof. The ideal (y,6) lies in some class, say a Thus there are two integers of K(u>), say ,fl2 different from zero such that ft y ,6 (s^ky.s) - ( 2^ k k^ that is (ftjY.Jljfi) - (fi2Yk'n26k^ Hence N(JJ1) N(Y,6) - N(fi2) N( Qtk) 39 or fll5l N( ^k* 5 B(ak> n2n2 " "<*•« Thus s ^nc Qir ) < a. k q where we let q > Max (/N( Qt .) ). lsiih 1 By Hurwitz' Lemma, there is a transformation T in L, a bl T c d such that That is, fljY - afi2yk + c fi26k ba dfi 6 OjY - 2Yk + 2 k * Set T • T. T. We then have o k y = aY, + c6. 'o k k 5 = bY, + d6, o k k an( t lus so il1y - ^2^0' ™ ^2®o * ^ YQ " °Y» 5q " 06 where |o| < q. q.e.d. ^ Lemma 3. To each (z,c) in H, there is a pair of integers (y>$) t (0,0) in K(w) for which the inequality dJ lw yyc 2 + (yz + 6)(yz + fi) i 72 holds. 40 Proof. We can write y = nx + n2io , y n j + n2w 6 • n3 + n^w , 6 113 + n^oi where nl, n2, n3, n^ are rational integers. Then yyc2 + (yz + fi) (yz + 6) =• yy(52 + ri2 + c2)'+ y^z + 6yz + 66 =• yyp2 + y6z + y^2 + 66 1' n2» n3» y (1) 0 0 i1 y 1 u 0 0 W n 6 0 0 1 a> « / \0 0 1 b) 1%/ and e. y z_ £l 0 0 y 2 2 yyp2 + y^z + y^z + 66 (y»y» = y t v,m y »sj *+ = jj* Wc M W n. The determinant of the quadratic form is _ .2 p - zz det (W M W) (w - ui) (u> - dv 16 By a result of Hermite with coefficient /2 due to Korkine and Zolotareff ( see Koksma (1936) §§17-18 ), we have a non-zero solution in nx, n2, n3, n4 for which 41 0 < YYC2 + (yz + Q • £ • D. Lemma 4. For each (z,0 in H, there is a Tq in L such that (z',C') = Tq(Z,C) with , where q is given by Lemma 2. Proof. For (z,C) one uses Lemma 3 to obtain y,S and then uses Lemma 2 to obtain T such that o c - — 5 rr^r- > : 5 i —• 2 Z + 2 2 + z + 2 YoYQt + ^0 ^ ^ q (YYC (Y fi) (yz + 6) q d Q.E.D. Note; If there is only one ideal class, then L = T and T is in T. /2 In this case there always is a point with Q £ which is equivalent to any given point, so the fundamental domain of T meets H - H at 00 only. We now go into the proof of Theorem 1. Consider the groups k -1 k T = r (k=l,...,h). The affine group of each of these is given by Lemma 1, and it consists of translations and rotations about vertical axes, so it maps the intersection of H with /2 CD ^ onto itself ( where e > 0 is sufficiently small ). Now take a funda- k mental domain for T bounded by planes and isometric spheres and take its intersection with (1) to obtain a region T^ F^. The union h F F. o " Uk»l k contains a point or has a boundary point equivalent to any given (z,0 in H under a transformation of T. To see this, determine T = T, T o k /2 with T in T by Lemma 4 such that (z, ,£.) « T (z,5) has > —jrr - e. i a 0 i q d 1c Now find (z2,C2) equivalent to (zj.C^) under in so that (z2,c2) is in T. F, . Then k k <*3.'3> " tk' *k tk tu'c) is in F^ and T"1 T^ T is in T. Since each has only one para bolic cusp, F(a>) can have no more that h parabolic cusps. We want to show that the points interior to Fq near any one of the parabolic cusps of the F^ are not equivalent under T. Then by h choosing in T, the points near the parabolic cusps of jL/ F^ will lie in F(u>). This will show that F(w) has at least h-inequivalent parabolic cusps. Let T^ denote the set of points of T^ Ffc such that t > t > 0. Now assume that a point (z,£) in T^ is equi valent to another point (z^,^) in F by means of a transformation a b in T. c d The point (z^,^) is in a region F^. Thus (z^Cj) - T(z,C). Let (z2,i;2) » T^(z,?) a = and (ZJJCJ) (z 1 * l ^ ^ (z2,C2). Then (z ,C ) is in T. so C, i t and * ' J J 1 1 Tk T T- (zj,,C2) - TkT- Denote the second row of T^ T T^1 by (y,6). Then d fi y - (afij - byj)yk + (cfij - yj) k- 43 Consider the two cases y = 0 and y M- If y = 0, then <^k'^k) ant^ ~ ~ ** (Yj.ij) .determine the same 1 s w ideal class so j=k and (z3,?3) = T Tk (z2>C2) i ^ ^ ith ™ 1 k * lc T, T T, in T . Since T. F, has no points equivalent under T kk« kk 00 are on unless they are on the boundary, the points (z3,S3) and (z2,£2) 1 s 1111 the boundary of p£. If y f 0, then because T T^ (z2,C2) i F^, it follows that /L - e < - 2 2 z + 6 q d YYC2 + (Y 2 >(YZ2 + 6) YY;2 On the other hand, { > t so — < We may choose t such that ^ sot d t > q . /2 - qzde to get a contradiction. Thus, the points interior to Fq near one of the h-parabolic cusps are not equivalent under T. Q.E.D. 2. The fundamental domain of r(u). It is easily seen that the only multipliers of are (i) ± 1, ± i when D = 1, i ±+ 1i + i/3 (ii) t 1, ^ when D «• 3, (iii) - 1 when D = 2 or D > 3. These give rise to the following rotations in ^(u): (i) when D » 1, 1 + i/3 - 1 + i/3 0 2 and D • 3, (ii) - 1 - i/3 1 - i/3 0 44 (iii) no rotations in ^(w) when D = 2 or D > 3. Figures 1, 2 and 3 give the fundamental domains for ^(ui) , where the boundary identifications are given by arrows. It is possible to give a plane representation for the fundamental domain as in Fig ures 1 to 5. Bianchi (1952) has determined the complete fundamental domains for all D £ 21. Let us do this by isometric spheres for the simplest cases. For the Picard group, the region exterior to all isometric spheres consists of the region exterior to all spheres with centers m + ni and radii 1. These spheres come from the transformations 'a b 1 - m - ni where a(- m - ni) - b = 1 with a and b Gaussian integers. The /2 smallest value of 5 for any point exterior to these spheres is — /z and the largest radius of any other isometric sphere is —. Thus the fundamental domain for the Picard group T(i) is given by Figure 4 where the arrow gives the new boundary identification. i i For the group T( ^ ), there are isometric spheres of radius 1 with centers at m + nto. The smallest value of £ in the re- /3 gion exterior to these spheres is — and the largest radius for any /2 1 + i/3 . other sphere is ~ , so the fundamental domain for T( ^ given by Figure 5, where the new boundary identification is given by the arrow. A similar argument shows that the fundamental domain for r(i/2) is given by Figure 6. / 1 1* * -» ' I / £ . _ ov / / s / Figure 1. The fundamental domain of r (i). Figure 2. The fundamental domain of r ( ^ *** \ «ov 2 46 Figure 3. The fundamental domain of ^(ui) where D = 2 or D > 3 "5 Figure 4. The fundamental domain of r(i). Figure 5. The fundamental domain of f( ), 48 f Figure 6. The fundamental domain of r(±/Z). 49 T Figure 7. The fundamental domain of r(i/5). 50 The fundamental domain for T(i/5) has two inequivalent singular points and is drawn in Figure 7 ( see Bianchi (1952) p.313 ). Some of the other fundamental domains are quite simple, for example T( ^ 1 i /if and T( • 2- )» while others are quite difficult to render in three dimensions, for example r(i/L0) and r(i/l3) ( see Bianchi (1952) pp.318, 327, 323 ). We shall give some of the topological properties of the funda mental domain for r(o>), namely, the topological structure at each point and the Euler characteristic. Definition. If T in F has a neighborhood in F which is homeomorphic to a 3-ball, then t will be called a manifold point of F. If no neighbor- nood of T in F is homeomorphic to a 3-ball, then T will be called a non-manifold point of F. It is clear that an interior point of the fundamental domain is a manifold point. The neighborhood of a point t of an ordinary edge e is made up of pieces which come from the different edges which belong to the same edge cycle as e. If the cycle is accidental, then the pieces form a complete neighborhood at x and thus T is a manifold point of F. If the cycle is of order n, then the pieces fit together to form an angular sector of angle ( see Figure 8 ). The two faces bound ing this sector are identified by an elliptic transformation of order n, and thus t is a manifold point of F. For a vertex v, there are several cases depending upon the nature of the cycle to which v belongs. However, the possibilities are limited by a simple result of Bianchi (1952) p.297. 51 Lemma 5. The only elliptic transformations in r(uj) are of orders (1) 2 and 3 if D = 1, (2) 2 and 4 if D = 2, (3) 2 , 3 and 6 if D = 3, (4) 2 and 3 if D > 3. This lemma limits the cycles to which v belongs to the follow ing ( eleven ) types: ( see Chapter III, section 2 ) (i) accidental, (ii) cyclic of orders 2, 3, 4 and 6, (iii) dihedral of orders 4, 6, 8 and 12, (iv) tetrahedral, (v) octahedral. There can be no icosahedral vertex cycles because there are no elliptic transformations of order five. Let us deal with each type of vertex _ycle separately. If v comes from an accidental vertex cycle, then the neighbor hoods in F of those vertices which are I"-equivalent to v can be mapped to v in such a way that they form a complete neighborhood of v in H. Choose a small neighborhood of v and take its intersection with each of the fundamental domains surrounding v. Now map these fundamental domains onto the fundamental domain F which contains v. This shows that the neighborhood of v in F is homeomorphic to a 3-ball, so v is a manifold point. If v belongs to an elliptic cycle, then the vertices which are T-equivalent to v can be mapped to v in such a way as to form an 52 elliptic sector as in Figure 8, and the elliptic transformation fixing v gives an identification which makes v a manifold point. If v belongs to a dihedral cycle, then the vertices which are T-equivalent to v can be mapped to form a dihedral sector as in Figure 9, and the group of elliptic transformations fixing v gives identifica tions which make v a manifold point. If v belongs to a tetrahedral cycle, then the vertices in F which are T-equivalent to v can be mapped to form a tetrahedral sector as in Figure 10, and the tetrahedral group of elliptic transformations fixing v gives identifications which make v a manifold point. If v belongs to an octahedral cycle, then the vertices in F which are T-equivalent to v can be mapped to form an octahedral sector as in Figure 11, and the octahedral group of elliptic transformations fixing v gives identifications which make v a manifold point. Thus the ordinary points of the fundamental domain are all manifold points. The only points left to consider are the singular points. The neighborhoods of these points depend upon the value of D. For D = 1 and D =• 3, Figures 12 and 13 show that 00 is a manifold point of the fundamental domain. When D ^ 1 or 3, Figure 14 shows that a singular point x is not a manifold point because the local homology groups at t are those of a torus. Thus we have the following theorem: 1 + i/3 Theorem 2. The fundamental domains for T(i) and T( - ) are manifolds, while for the other groups of Bianchi, the fundamental domains are pseudomanifolds which are manifolds at every point except the singular vertices. The homology groups at the singular points of the non-manifolds are those of a torus. Figure 8. The angular sector at an edge e. Figure 9. The sector at a dihedral vertex v. 54 Figure 10. The sector at a tetrahedral vertex v Figure 11. The sector at an octahedral vertex v. Figure 12. The neighborhood of 00 for T(i). co Figure 13. The neighborhood of 00 for r( 14. The neighborhood of a singular vertex for r(ui) where D f 1 or 3. It is clear that the fundamental domains are triangulable and in fact are polyhedra. Such a three dimensional complex which arises from a pairwise identification of faces is a manifold if and only if its Euler characteristic x is zero ( see Seifert - Threlfall (1934) §60 ). Since the Jacobian of the transformations is always positive, the fundamental domain is orientable. Consider a polyhedron which is a manifold except that at certain vertices the boundary of every small neighborhood of the vertex is some orientable surface. Suppose v^ is a vertex whose neighborhoods have a boundary which is a surface of genus p^ ( for the groups of Bianchi, p » 1 ). Cut all of these points v^ out by small neighbor hoods and form the double of the resulting manifold with boundary. Removing a point lowers the characteristic x by 1 while the surface S^ at v. contributes xQ = 2 - 2p. to the characteristic. The double x j, region is a manifold and hence = Thus h h h 0 = x ~ n + 2n - 2 £ p i=l h or x - I (2p. - 1), i=l where h is the number of non-manifold points. Theorem 3. Let h be the class number of K(w). Then 1 + iv C 0 for T(i), r( 2 ^ ) * v- h for r(ui) when D / 1 or 3. 57 1 , . /r For the simplest groups T(i), T(i/2) and T( 2^— ) •"-s easy to determine further topological properties. The fundamental domain for T(i) is homeomorphic to half space H with the identifica tions of the boundary shown in Figure 15. Making these identifica tions gives Figure 16. The final identification makes the fundamental domain for T(i) homeomorphic to the sphere S3. The homology groups 3 for S are Hq = Z, H1 • H2 =» 0, H3 = Z. In a similar manner Figure 17 shows that the fundamental domain 1 "f" i^ 3 for T( 2 ) *-s homeomorphic to the sphere S3. The fundamental domain for T(i/2) can be changed by a cut and paste operation to the region in Figure 18. Using the orientations as given in that figure, we get the following block incidence matrices ( see Seifert - Threlfall (1934) §22 ) which give the homology groups Hq - Z, Hx - 0, H2 = Z, H3 = Z. * O E° Q2 Q3 0 0 0 E1 Q? <*2 E2 QI I 0 0 -1 Q? 0 Q2 0 1 -1 0 ^2 0 Q3 0 0 0 -1 Q23 0 QS 0 f, S , f, f, ^ Figure 15. The initial identification in r(i) Figure 16. The final identification in T(i) a. ""K f. «^"v l + l/J Figure 17. The identifications in T( 59 <*? Figure 18. The fundamental domain of T(i/2) as a polyhedron. 60 3. Conclusion. The preceding two sections lay the foundation for the study of the fundamental domains of the groups of Bianchi. The next goal should be a further study of the topology of the fundamental domains of r(u>) for all 10, in particular, the homology groups and the fundamental group irn. After the properties of the fundamental domains of T (oi) are found one can turn to the subgroups of r(u>), in particular, the princi pal congruence subgroups modulo an ideal. These normal subgroups give rise to superdomains whose topological structure can be investigated. Sansone (1923) has done work in this direction for the groups T(i) and "f" i/j F( Since the number of fundamental domains of r(ai) which make up the fundamental domain for a principal congruence subgroup becomes large with the norm of the ideal, the study of these domains is difficult. It would be desirable to give a theory analogous to the plane theory where the singular points are a new factor. Finally, since the fundamental domains are not manifolds in general, it would be desirable to see how much of the topological theory of manifolds can be extended to these pseudomanifolds. REFERENCES Bianchi, L. Opere Matematiche, Vol.l,pt.l, Roma, 1952. Caratheodory, C. Theory of functions of a complex variable. Vol.1, Chelsea, New York, 1954. Fatou, P. Fonctions automorphes, Vol.2 of Tb^orie des fonctions alg^briques... , P.E. Appell and £. Goursat, Gauthiers - Villars, Paris, 1930. Ford, L.R. Automorphic functions, McGraw Hill, New York, 1929. Fricke, R. and Klein, F. Vorlesungen uber die Theorie der Automorphen Funk- tionen, Vol.1, Teubner, Leipzig, 1897. Koksma, J.F. Diophantische Approximationen, Springer, Berlin, 1936. Lehner, J. Discontinuous groups and automorphic functions, Amer. Math. Soc.,Providence, 1964. Maass. H. Uber Gruppen von hyperabelschen Transformationen. Sitzber. Heidelberg Akad. Wiss., 1940. Poincare, H. Sur les groupes Kleingens, Acta. Math., 3, 1883. Sansone,* G. l I sottogruppi del Gruppo di Picard e due Teoremi sui Gruppi finiti analoghi al Teorema del Dyck, Rend. Circ. Math., Palermo, 47, 1923. Seifert, H. and Threlfall, W. Lehrbuch der Topologie, Teubner, Leipzig, 1934. 61