Finite Presentability of S-Arithmetic Groups Compact Presentability of Solvable Groups
The problem of determining which S-arithmetic groups have a finite presentation is solved for arbitrary linear algebraic groups over finite extension fields of #3. For certain solvable topological groups this problem may be reduced to an analogous problem
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1261 Herbert Abels
Finite Presentability of S-Arithmetic Groups Compact Presentability of Solvable Groups
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1261 Herbert Abels
Finite Presentability of S-Arithmetic Groups Compact Presentability of Solvable Groups
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Herbert Abels Universitat Bielefeld, Fakultat fur Mathematik 4800 Bielefeld, Federal Republic of Germany
Mathematics Subject Classification (1980): Primary: 20F05, 20G30, 20G25, 22D05, 20F 16 Secondary: 20F 18, 20G 10, 20G35, 11 E99, 17B55, 20J05, 22E99 ISBN 3-540-17975-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387 -17975-5 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
Fur Elisabeth
CONTENTS
O.
I.
Introduction 0.1
Abstract
1
0.2
The problem and its solution
1
0.3
Survey of the proof, contents of the chapters
8
0.4
More general questions
11
0.5
Acknowledgements . .
13
.
Compact presentability and contracting automorphisms
15 15
1.1
Compact presentability
1.2
Contracting automorphisms
17
1.3
Compact presentability and contracting automorphisms
23
II. Filtrations of Lie algebras and groups
27
2.1
Filtered Lie algebras.
27
2.2
Formulae on commutators
30
2.3
Filtered groups.
31
.
. .
2.4
Some results about nilpotent groups
33
2.5
Nilpotent Lie groups over fields of characteristic zero
36
2.6
Nilpotent Lie groups over p-adic fields.
44
.
III. A necessary condition for compact presentability . .
.
.
. .
49
3.1
Weights.
.
50
3.2
Values of weights
52
3.3
A counterexample
56
3.4
The geometric invariant of Bieri and Strebel
59 61
IV. Implications of the necessary condition 4.1 The building blocks NC 4.2
The colimits
4.3
Commutators in
Hand
4.4
The descending central series of
62 64
M
68
n M
71
VI
4.5 4.6 4.7 V.
80
H changed into a top logical group H U The kernel of H Q N U The colimit of the Lie algebras n C
The second homology . . .
83
87 .
90
.
93
5. 1
Homology of a group. The Hopf extension
5.2
Homology of a Lie algebra
5.3
Lie algebra homology versus group homology
5.4
A topology on the Hopf extension
102
5.5
107
5.6
H H 2(nlmp)o 2(n!K)o The main theorem
111
5.7
Examples
115
VI. S-arithmetic groups
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