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University Microfilms, Inc., Ann Arbor, Michigan the SINGULAR POINTS of the FUNDAMENTAL THE SINGULAR POINTS OF THE FUNDAMENTAL DOMAINS FOR THE GROUPS OF BIANCHI Item Type text; Dissertation-Reproduction (electronic) Authors Woodruff, William Munger, 1936- Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 27/09/2021 03:02:59 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 <j> defined on r whose values are functions $(T): X -*• X such that (i) <KT): X + X is a homeomorphisra of X for every T in r, (ii) (S)°(f>(T) «* <|»(ST) for every S and T in r. Usually, one omits <J> and thinks of T as acting directly on an element x of X; namely, T(x) • [<|>(T)](x). Thus, property (ii) is just 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.
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