Profinite Groups

by Ribes, Luis; Feferman, S.; Gromov, M.; Jost, J.; Kollar, J.; Lenstra, H. W., Jr.; Lions, P. L.

• ISBN13: 9783540669869
• ISBN10: 3540669868
• Format: Hardcover
• Copyright: 2000-11-01
• Publisher: Springer Verlag

Summary

The aim of this book is to serve both as an introduction to profinite groups and as a reference for specialists in some areas of the theory. The book is reasonably self-contained. Profinite groups are Galois groups. As such they are of interest in algebraic number theory. Much of recent research on abstract infinite groups is related to profinite groups because residually finite groups are naturally embedded in a profinite group. In addition to basic facts about general profinite groups, the book emphasizes free constructions (particularly free profinite groups and the structure of their subgroups). Homology and cohomology is described with a minimum of prerequisites.

Table of Contents

Preface	vii
Inverse and Direct Limits
Inverse or Projective Limits	1	(13)
Direct or Inductive Limits	14	(4)
Notes, Comments and Further Reading	18	(1)
Profinite Groups
Pro - C Groups	19	(9)
Basic Properties of Pro - C Groups	28	(4)
Existence of Sections	29	(2)
Exactness of Inverse Limits of Profinite Groups	31	(1)
The Order of a Profinite Group and Sylow Subgroups	32	(11)
Generators	43	(2)
Finitely Generated Profinite Groups	45	(3)
Generators and Chains of Subgroups	48	(4)
Procyclic Groups	52	(2)
The Frattini Subgroup of a Profinite Group	54	(6)
Pontryagin Duality for Profinite Groups	60	(8)
Pullbacks and Pushouts	68	(2)
Profinite Groups as Galois Groups	70	(5)
Notes, Comments and Further Reading	75	(4)
Free Profinite Groups
Profinite Topologies	79	(3)
The Pro - C Completion	82	(8)
The Completion Functor	85	(5)
Free Pro - C Groups	90	(11)
Free Pro - C Group on a Set Converging to 1	93	(8)
Maximal Pro - C Quotient Groups	101	(2)
Characterization of Free Pro - C Groups	103	(14)
Open Subgroups of Free Pro - C Groups	117	(3)
Notes, Comments and Further Reading	120	(3)
Some Special Profinite Groups
Powers of Elements with Exponents from Z	123	(1)
Subgroups of Finite Index in a Profinite Group	124	(9)
Profinite Abelian Groups	133	(3)
Automorphism Group of a Profinite Group	136	(7)
Automorphism Group of a Free Pro-p Group	143	(4)
Profinite Frobenius Groups	147	(7)
Torsion in the Profinite Completion of a Group	154	(6)
Notes, Comments and Further Reading	160	(5)
Discrete and Profinite Modules
Profinite Rings and Modules	165	(7)
Duality Between Discrete and Profinite Modules	171	(1)
Free Profinite Modules	172	(3)
G-modules and Complete Group Algebras	175	(4)
The Complete Group Algebra	177	(2)
Projective and Injective Modules	179	(4)
Complete Tensor Products	183	(4)
Profinite G-spaces	187	(8)
Free Profinite [[RG]]-modules	195	(1)
Diagonal Actions	196	(3)
Notes, Comments and Further Reading	199	(2)
Homology and Cohomology of Profinite Groups
Review of Homological Algebra	201	(9)
Right and Left Derived Functors	205	(1)
Bifunctors	206	(1)
The Ext Functors	207	(1)
The Tor Functors	208	(2)
Cohomology with Coefficients in DMod([RG])	210	(4)
Standard Resolutions	211	(3)
Homology with Coefficients in PMod([RG])	214	(5)
Cohomology Groups with Coefficients in DMod(G)	219	(1)
The Functorial Behavior of Hn(G, A) and Hn(G, A)	220	(7)
The Inflation Map	221	(6)
Hn(G, A) as Derived Functors on DMod(G)	227	(4)
Special Mappings	231	(7)
The Restriction Map in Cohomology	231	(1)
The Corestriction Map in Cohomology	232	(4)
The Corestriction Map in Homology	236	(1)
The Restriction Map in Homology	236	(2)
Homology and Cohomology Groups in Low Dimensions	238	(7)
H2(G, A) and Extensions of Profinite Groups	240	(5)
Extensions of Profinite Groups with Abelian Kernel	245	(5)
Induced and Coinduced Modules	250	(5)
The Induced Module IndGH(B) for H Open	255	(2)
Notes, Comments and Further Reading	257	(2)
Cohomological Dimension
Basic Properties of Dimension	259	(5)
The Lyndon-Hochschild-Serre Spectral Sequence	264	(5)
Cohomological Dimension of Subgroups	269	(5)
Cohomological Dimension of Normal Subgroups and Quotients	274	(2)
Groups G with cdp(G) ≤ 1	276	(4)
Projective Profinite Groups	280	(4)
Free Pro-p Groups and Cohomological Dimension	284	(3)
Generators and Relators for Pro-p Groups	287	(4)
Cup Products	291	(7)
Notes, Comments and Further Reading	298	(4)
Normal Subgroups of Free Pro - C Groups
Normal Subgroup Generated by a Subset of a Basis	302	(2)
The S-rank	304	(6)
Accessible Subgroups	310	(5)
Accessible Subgroups H with wo(F / H) < rank(F)	315	(7)
Homogeneous Pro - C Groups	322	(13)
Normal Subgroups	335	(9)
Proper Open Subgroups of Normal Subgroups	344	(5)
The Congruence Kernel of SL2(Z)	349	(1)
Sufficient Conditions for Freeness	350	(7)
Characteristic Subgroups of Free Pro - C Groups	357	(3)
Notes, Comments and Further Reading	360	(1)
Free Constructions of Profinite Groups
Free Pro - C Products	361	(14)
Amalgamated Free Pro - C Products	375	(8)
Cohomological Characterizations of Amalgamated Products	383	(7)
Pro - C HNN-extensions	390	(7)
Notes, Comments and Further Reading	397	(4)
Open Questions	401	(14)
Appendix
A1 Spectral Sequences	403	(2)
A2 Positive Spectral Sequences	405	(4)
A3 Spectral Sequence of a Filtered Complex	409	(2)
A4 Spectral Sequences of a Double Complex	411	(2)
A5 Notes, Comments and Further Reading	413	(2)
Bibliography	415	(10)
Index of Symbols	425	(4)
Index of Authors	429	(2)
Index of Terms	431