The Arithmetic of Hyperbolic 3-Manifolds

by MacLachlan, Colin; Reid, Alan W.

• ISBN13: 9780387983868
• ISBN10: 0387983864
• Format: Hardcover
• Copyright: 2002-11-01
• Publisher: Springer Nature

Summary

For the past 25 years, the Geometrization Program of Thurston has been a driving force for research in 3-manifold topology. This has inspired a surge of activity investigating hyperbolic 3-manifolds (and Kleinian groups), as these manifolds form the largest and least well-understood class of compact 3-manifolds. Familiar and new tools from diverse areas of mathematics have been utilized in these investigations, from topology, geometry, analysis, group theory, and from the point of view of this book, algebra and number theory. Thisbook is aimed at readers already familiar with the basics of hyperbolic 3-manifolds or Kleinian groups, and it is intended to introduce them to the interesting connections with number theory and the tools that will be required to pursue them. While there are a number of texts which cover the topological, geometric and analytical aspects of hyperbolic 3-manifolds, this book is unique in that it deals exclusively with the arithmetic aspects, which are not covered in other texts. Colin Maclachlan is a Reader in the Department of Mathematical Sciences at the University of Aberdeen in Scotland where he has served since 1968. He is a former President of the Edinburgh Mathematical Society. Alan Reid is a Professor in the Department of Mathematics at The University of Texas at Austin. He is a former Royal Society University Research Fellow, Alfred P. Sloan Fellow and winner of the Sir Edmund Whittaker Prize from The Edinburgh Mathematical Society. Both authors have published extensively in the general area of discrete groups, hyperbolic manifolds and low-dimensional topology.

Table of Contents

Preface	v
Noumber-Theoretic Menagerie	1	(46)
Number Fields and Field Extensions	2	(4)
Algebraic Integers	6	(5)
Ideals in Rings of Integers	11	(9)
Units	20	(2)
Class Groups	22	(2)
Valuations	24	(5)
Completions	29	(6)
Adeles and Ideles	35	(4)
Quadratic Forms	39	(8)
Kleinian Groups and Hyperbolic Manifolds	47	(30)
PSL(2, C) and Hyperbolic 3-Space	47	(3)
Subgroups of PSL(2, C)	50	(5)
Hyperbolic Manifolds and Orbifolds	55	(2)
Examples	57	(5)
Bianchi Groups	58	(1)
Coxeter Groups	59	(1)
Figure 8 Knot Complement	59	(1)
Hyperbolic Manifolds by Gluing	60	(2)
3-Manifold Topology and Dehn Surgery	62	(5)
3-Manifolds	63	(1)
Hyperbolic Manifolds	64	(1)
Dehn Surgery	65	(2)
Rigidity	67	(2)
Volumes and Ideal Tetrahedra	69	(2)
Further Reading	71	(6)
Quaternion Algebras I	77	(34)
Quaternion Algebras	77	(5)
Orders in Quaternion Algebras	82	(5)
Quaternion Algebras and Quadratic Forms	87	(4)
Orthogonal Groups	91	(1)
Quaternion Algebras over the Reals	92	(2)
Quaternion Algebras over P-adic Fields	94	(4)
Quaternion Algebras over Number Fields	98	(3)
Central Simple Algebras	101	(4)
The Skolem Noether Theorem	105	(3)
Further Reading	108	(3)
Invariant Trace Fields	111	(22)
Trace Fields for Kleinian Groups of Finite Covolume	111	(3)
Quaternion Algebras for Subgroups of SL(2, C)	114	(2)
Invariant Trace Fields and Quaternion Algebras	116	(4)
Trace Relations	120	(3)
Generators for Trace Fields	123	(5)
Generators for Invariant Quaternion Algebras	128	(2)
Further Reading	130	(3)
Examples	133	(32)
Bianchi Groups	133	(1)
Knot and Link Complements	134	(1)
Hyperbolic Fibre Bundles	135	(2)
Figure 8 Knot Complement	137	(3)
Group Presentation	137	(1)
Ideal Tetrahedra	137	(1)
Once-Punctured Torus Bundle	138	(2)
Two-Bridge Knots and Links	140	(2)
Once-Punctured Torus Bundles	142	(1)
Polyhedral Groups	143	(9)
Non-compact Tetrahedra	144	(2)
Compact Tetrahedra	146	(3)
Prisms and Non-integral Traces	149	(3)
Dehn Surgery Examples	152	(7)
Jørgensen's Compact Fibre Bundles	152	(1)
Fibonacci Manifolds	153	(3)
The Weeks-Matveev-Fomenko Manifold	156	(3)
Fuchsian Groups	159	(3)
Further Reading	162	(3)
Applications	165	(32)
Discreteness Criteria	165	(3)
Bass's Theorem	168	(5)
Tree of SL(2, Kp)	169	(1)
Non-integral Traces	170	(1)
Free Product with Amalgamation	171	(2)
Geodesices and Totally Geodesic Surfaces	173	(7)
Manifolds with No Geodesic Surfaces	173	(1)
Embedding Geodesic Surfaces	174	(2)
The Non-cocompact Case	176	(2)
Simple Geodesics	178	(2)
Further Hilbert Symbol Obstructions	180	(3)
Geometric Interpretation of the Invariant Trace Field	183	(6)
Constructing Invariant Trace Fields	189	(5)
Further Reading	194	(3)
Orders in Quaternion Algebras	197	(28)
Integers, Ideals and Orders	197	(3)
Localisation	200	(5)
Discriminants	205	(2)
The Local Case-I	207	(2)
The Local Case - II	209	(5)
Orders in the Global Case	214	(3)
The Type Number of a Quaternion Algebra	217	(6)
Further Reading	223	(2)
Quaternion Algebras II	225	(28)
Adeles and Ideles	226	(3)
Duality	229	(4)
Classification of Quaternion Algebras	233	(4)
Theorem on Norms	237	(1)
Local Tamagawa Measures	238	(6)
Tamagawa Numbers	244	(2)
The Strong Approximation Theorem	246	(4)
Further Reading	250	(3)
Arithmetic Kleinian Groups	253	(22)
Discrete Groups from Orders in Quaternion Algebras	254	(3)
Arithmetic Kleinian Groups	257	(4)
The Identification Theorem	261	(6)
Complete Commensurability Invariants	267	(5)
Algebraic Integers and Orders	272	(2)
Further Reading	274	(1)
Arithmetic Hyperbolic 3-Manifolds and Orbifolds	275	(30)
Bianchi Groups	275	(2)
Arithmetic Link Complements	277	(4)
Zimmert Sets and Cuspidal Cohomology	281	(4)
The Arithmetic Knot	285	(2)
Fuchsian Subgroups of Arithmetic Kleinian Groups	287	(5)
Fuchsian Subgroups of Bianchi Groups and Applications	292	(5)
Simple Geodesics	297	(2)
Hoovering Up	299	(3)
The Finite Subgroups A4, S4 and A5	299	(1)
Week's Manifold Again	300	(2)
Further Reading	302	(3)
Discrete Arithmetic Groups	305	(26)
Orthogonal Groups	306	(4)
SO(3, 1) and SO(2, 1)	310	(5)
General Discrete Arithmetic Groups and Margulis Theorem	315	(7)
Reflection Groups	322	(7)
Arithmetic Polyhedral Groups	325	(1)
Tetrahedral Groups	326	(1)
Prismatic Examples	327	(2)
Further Reading	329	(2)
Commensurable Arithmetic Groups and Volumes	331	(40)
Covolumes for Maximal Orders	332	(6)
Consequences of the Volume Formula	338	(7)
Arithmetic Kleinian Groups with Bounded Covolume	338	(2)
Volumes for Eichler Orders	340	(1)
Arithmetic Manifolds of Equal Volume	341	(1)
Estimating Volumes	342	(1)
A Tetrahedral Group	343	(2)
Fuchsian Groups	345	(7)
Arithmetic Kleinian Groups with Bounded Covolume	345	(1)
Totally Real Fields	346	(1)
Fuchsian Triangle Groups	346	(4)
Signatures of Arithmetic Fuchsian Groups	350	(2)
Maximal Discrete Groups	352	(4)
Distribution of Volumes	356	(2)
Minimal Covolume	358	(5)
Minimum Covolume Groups	363	(5)
Further Reading	368	(3)
Length and Torsion in Arithmetic Hyperbolic Orbifolds	371	(44)
Loxodromic Elements and Geodesics	371	(2)
Geodesics and Embeddings in Quaternion Algebras	373	(4)
Short Geodesies, Lehmer's and Salem's Conjectures	377	(6)
Isospectrality	383	(11)
Torsion in Arithmetic Kleinian Groups	394	(11)
Volume Calculations Again	405	(5)
Volumes of Non-arithmetic Manifolds	410	(3)
Further Reading	413	(2)
Appendices	415	(28)
Compact Hyperbolic Tetrahedra	415	(1)
Non-compact Hyperbolic Tetrahedra	416	(2)
Arithmetic Groups	416	(1)
Non-arithmetic Groups	417	(1)
Arithmetic Fuchsian Triangle Groups	418	(1)
Hyperbolic Knot Complements	419	(4)
Small Closed Manifolds	423	(8)
Small Cusped Manifolds	431	(5)
Arithmetic Zoo	436	(7)
Non-compact Examples	436	(3)
Compact Examples, Degree 2 Fields	439	(1)
Compact Examples, Degree 3 Fields	440	(1)
Compact Examples, Degree 4 Fields	441	(2)
Bibliography	443	(16)
Index	459