Discontinuous groups of isometries in the hyperbolic plane:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
<<de>> Gruyter
2003
|
| Schriftenreihe: | De Gruyter studies in mathematics
29 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 355 - 357 |
| Beschreibung: | XXI, 364 S. Ill., graph. Darst. |
| ISBN: | 3110175266 |
Internformat
MARC
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| 100 | 1 | |a Fenchel, Werner |d 1905-1988 |e Verfasser |0 (DE-588)117709980 |4 aut | |
| 245 | 1 | 0 | |a Discontinuous groups of isometries in the hyperbolic plane |c Werner Fenchel ; Jakob Nielsen. Ed. by Asmus L. Schmidt |
| 264 | 1 | |a Berlin [u.a.] |b <<de>> Gruyter |c 2003 | |
| 300 | |a XXI, 364 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a De Gruyter studies in mathematics |v 29 | |
| 500 | |a Literaturverz. S. 355 - 357 | ||
| 650 | 4 | |a Groupes discontinus | |
| 650 | 7 | |a Groupes discontinus |2 ram | |
| 650 | 4 | |a Isométrie (Mathématiques) | |
| 650 | 7 | |a Isométrie (mathématiques) |2 ram | |
| 650 | 4 | |a Discontinuous groups | |
| 650 | 4 | |a Isometrics (Mathematics) | |
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| 650 | 0 | 7 | |a Riemannsche Fläche |0 (DE-588)4049991-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hyperbolische Geometrie |0 (DE-588)4161041-6 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Nielsen, Jakob |d 1890-1959 |e Verfasser |0 (DE-588)117730459 |4 aut | |
| 700 | 1 | |a Schmidt, Asmus L. |e Sonstige |4 oth | |
| 830 | 0 | |a De Gruyter studies in mathematics |v 29 |w (DE-604)BV000005407 |9 29 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-010170003 | ||
Datensatz im Suchindex
| _version_ | 1804129781639282688 |
|---|---|
| adam_text | Contents
Preface v
Life and work of the Authors by Bent Fuglede xv
I Mobius transformations and non euclidean geometry 1
§1 Pencils of circles inversive geometry 1
1.1 Notations 1
1.2 Three kinds of pencils 1
1.3 Determination of pencils 2
1.4 Inverse points 3
§2 Cross ratio 4
2.1 Definition and identities 4
2.2 Amplitude and modulus 4
2.3 Harmonic pairs 5
§3 Mobius transformations, direct and reversed 6
3.1 Invariance of the cross ratio 6
3.2 Determination by three points 7
3.3 Reversed transformations 7
3.4 Inversions 8
§4 Invariant points and classification of Mobius transformations 8
4.1 The multiplier 8
4.2 Two invariant points 9
4.3 One invariant point 10
4.4 Transformations with an invariant circle 11
4.5 Permutable transformations 12
§5 Complex distance of two pairs of points 14
5.1 Definition 14
5.2 Relations between distances 16
§6 Non euclidean metric 18
6.1 Terminology of non euclidean geometry 18
6.2 Distance and angle 20
§7 Isometric transformations 23
7.1 Motions and reversions 23
7.2 Classification of motions and reversions 24
7.3 Products of reflections and half turns 25
7.4 Transforms 26
§8 Non euclidean trigonometry 27
8.1 The special trigonometric formulae 27
8.2 Properties of non euclidean metric 31
8.3 Area 32
8.4 Sine amplitude 34
8.5 Projection of lines 34
8.6 Separation of collections of lines 35
8.7 The general trigonometric formulae 37
§9 Products and commutators of motions 43
9.1 Three motions with product 1 43
9.2 Composition by half turns 44
9.3 Composition by reflections 45
9.4 Calculation of invariants in the symmetric case 50
9.5 Relations for three commutators 50
9.6 Foot triangle 51
9.7 Circumscribed trilateral 53
9.8 Invariant of commutators 55
9.9 Some auxiliary results 56
II Discontinuous groups of motions and reversions 58
§10 The concept of discontinuity 58
10.1 Some notations and definitions 58
10.2 Discontinuity in £) 58
10.3 The distance function 60
10.4 Regular and singular points 60
10.5 Fundamental domains and fundamental polygons 61
10.6 Generation of S 63
10.7 Relations for 3 63
10.8 Normal domains 68
§ 11 Groups with invariant points or lines 70
11.1 Quasi abelian groups 70
11.2 Groups with a proper invariant point 70
11.3 Groups with an infinite invariant point 71
11.4 Non discontinuous groups with an infinite invariant point ... 73
11.5 Groups with an invariant line 74
11.6 List of quasi abelian groups 77
11.7 Conclusion 77
§ 12 A discontinuity theorem 78
12.1 An auxiliary theorem 78
12.2 The discontinuity theorem 79
12.3 Proof for groups containing rotations 80
12.4 Proof for groups not containing rotations 80
§ 13 {^ groups. Fundamental set and limit set 82
13.1 Quasi abelian subgroups of 5 82
13.2 Equivalence classes with respect to 5 82
13.3 The fundamental set g( $) 84
13.4 The limit set g(£) 85
13.5 Accumulation in limit points 86
13.6 A theorem on sequences of elements of J 87
13.7 Accumulation points for equivalence classes 93
13.8 Fundamental sequences 94
§ 14 The convex domain of an 5 group. Characteristic and isometric neigh¬
bourhood 95
14.1 The convex domain X (#) 95
14.2 Boundary axes and limit sides 97
14.3 Further properties of the convex domain 98
14.4 Characteristic neighbourhood of a point in £ 100
14.5 Distance modulo J 101
14.6 Isometric neighbourhood of a point in D 101
14.7 Characteristic neighbourhood of a limit centre 105
14.8 Isometric neighbourhood of a limit centre 107
14.9 Centre free part of X (£) 112
14.10 Truncated domain of an J group 113
§15 Quasi compactness modulo finite generation of £ 115
15.1 Quasi compactness and compactness modulo $ 115
15.2 Some consequences of quasi compactness 117
15.3 Quasi compactness and finite generation 119
15.4 Generation by translations and reversed translations 120
15.5 Necessity of the condition in the main theorem of Section 3 .122
15.6 The hull of a finitely generated subgroup 126
III Surfaces associated with discontinuous groups 127
§16 The surfaces £) modulo Oand JC (£) modulo 5 127
16.1 The surface £ mod 0 127
16.2 Surfaces derived from quasi abelian groups 128
16.3 Geodesies 129
16.4 Description of £ mod 5 131
16.5 Reflection chains and reflection rings 132
16.6 The surface JC(5) mod 5 133
16.7 The surface X*(£) mod 3 134
§17 Area and type numbers 135
17.1 Properties of normal domains 135
17.2 Normal domains in the case of quasi compactness 139
17.3 Area of X (£) mod # 141
17.4 Type numbers 144
17.5 Orientability 146
17.6 Characteristic and genus 147
17.7 Relation between area and type numbers 149
IV Decompositions of groups 153
§18 Composition of groups 153
18.1 Generalized free products 153
18.2 Generalized free product of two groups operating on two mu¬
tually adjacent regions 154
18.3 Properties of generalized free products 160
18.4 Quasi abelian groups as generalized free products 161
18.5 Tesselation of £ by the collection of domains 162
18.6 Surface corresponding to a generalized free product 163
18.7 Generalized free product of a group operating on a region with
a quasi abelian group operating on a boundary of that region . 164
18.8 Abstract characterization of the generalized free product . . .165
18.9 Surface corresponding to the generalized free product 167
18.10 Quasi abelian generalized free products 168
18.11 Generalized free product of infinitely many groups operating
on congruent regions 168
18.12 Extension of a group by an adjunction 173
§ 19 Decomposition of groups 174
19.1 Decomposition of an JJ group 174
19.2 Decomposition of 5 in the case I 177
19.3 Decomposition of 5 in the case II 179
19.4 Decomposition of 5 in the case III 180
19.5 Effect on the surface 180
19.6 Orientation of the decomposing line 182
19.7 Simultaneous decomposition 183
19.8 Decomposition by non dividing lines 186
19.9 Decomposition by dividing lines 187
19.10 L and # as generalized free products 191
§20 Decompositions of ^ groups containing reflections 196
20.1 The reflection subgroup 9t 196
20.2 A fundamental domain for H 197
20.3 Abstract presentation of the reflection group 199
20.4 Reflection chains and reflection rings 199
20.5 The finite polygonal disc 200
20.6 The polygonal cone 202
20.7 The infinite polygonal disc 202
20.8 The polygonal mast 203
20.9 Open boundary chain with different end points 203
20.10 The case of the full reflection line 204
20.11 The incomplete reflection strip 204
20.12 The crown 205
20.13 The complete reflection strip 206
20.14 The reflection crown 206
20.15 The conical reflection strip 206
20.16 The conical crown 207
20.17 The cross cap crown 207
20.18 The general case 207
20.19 The double reflection strip 211
20.20 The double crown 211
20.21 Determination of 5 by a free product with amalgamation . . .211
§21 Elementary groups and elementary surfaces 213
21.1 Two lemmas 213
21.2 Decomposition of £ by 54 215
21.3 Boundary chains 216
21.4 Reversibility and non reversibility 217
21.5 Rotation twins 217
21.6 Non reversibility of % 218
21.7 The case of a motion 219
21.8 Equivalence or non equivalence of P and 3 222
21.9 Elementary groups 222
21.10 Equal effect of 5 and £ in the interior of K( £) 224
21.11 Reversibility of S 224
21.12 Coincidence of the cases of reversibility and non reversibility 226
21.13 Elementary surfaces 227
21.14 The role of £ in the determination of £ 228
21.15 The handle and the Klein bottle 228
21.16 The case of a reversion 230
21.17 Metric quantities of elementary groups 233
§22 Complete decomposition and normal form in the case
of quasi compactness 242
22.1 Decomposition based on two protrusions 242
22.2 Existence of simple axes of reversed translations 245
22.3 Reduction of the genus in the case of non orientability .... 245
22.4 Reduction of the genus in the case of orientability 250
22.5 Existence of protrusions 256
22.6 Reduction by rotation twins 257
22.7 Further reduction based on protrusions 258
22.8 A characteristic property of elementary groups 261
22.9 Normal forms 262
22.10 Groups containing reflections 265
22.11 Normal form embracing all finitely generated ^ groups .... 268
§23 Exhaustion in the case of non quasi compactness 270
23.1 Decomposition by a subgroup 270
23.2 The extended hull of a subgroup 270
23.3 Exhaustion by extended hulls 273
23.4 Coverage of points of JC(5) 275
23.5 Coverage of points of 8. Ends of JC (£) and of X (3) mod $ 275
23.6 The kernel of a given subgroup 277
23.7 Exhaustion by kernels 280
V Isomorphism and homeomorphism 283
§24 Topological and geometrical isomorphism 283
24.1 Topological and geometrical isomorphism 283
24.2 Correspondence on S and S 286
24.3 Correspondence of extended hulls and of ends of the convex
domain 286
24.4 Correspondence of reflection lines 288
24.5 Relative location of corresponding centres and corresponding
lines of reflection 290
24.6 Correspondence of reflection chains and reflection rings . . . 293
24.7 Relative location of corresponding centres and corresponding
inner axes 294
24.8 Another characterization of t isomorphisms, applicable in the
case of quasi compactness 296
24.9 Remarks concerning the preceding section 306
24.10 Invariance of type numbers 307
§25 Topological and geometrical homeomorphism 308
25.1 / mappings 308
25.2 Extension of a f mapping to the boundary circles 312
25.3 g mappings 314
25.4 f homeomorphism and ^ homeomorphism 317
§26 Construction of g mappings. Metric parameters. Congruent groups .318
26.1 A lemma 318
26.2 g mappings of elementary groups 318
26.3 Metric parameters of elementary groups 321
26.4 Congruence of elementary groups 323
26.5 g mappings of handle groups 324
26.6 Metric parameters and congruence of handle groups 327
26.7 Finitely generated groups of motions with p 0 328
26.8 Finitely generated groups # without reflections and without
modulo SJ simple, non dividing axes 330
26.9 Finitely generated ^ groups without reflections 331
26.10 Finitely generated J groups containing reflections 333
26.11 g mappings of infinitely generated groups 340
26.12 Congruence of ^ groups. Alignment lengths 341
Symbols and definitions 349
Alphabets 353
Bibliography 355
Index 361
|
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| author | Fenchel, Werner 1905-1988 Nielsen, Jakob 1890-1959 |
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| dewey-sort | 3514 12 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV016447092 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:10:36Z |
| institution | BVB |
| isbn | 3110175266 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010170003 |
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| physical | XXI, 364 S. Ill., graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | <<de>> Gruyter |
| record_format | marc |
| series | De Gruyter studies in mathematics |
| series2 | De Gruyter studies in mathematics |
| spelling | Fenchel, Werner 1905-1988 Verfasser (DE-588)117709980 aut Discontinuous groups of isometries in the hyperbolic plane Werner Fenchel ; Jakob Nielsen. Ed. by Asmus L. Schmidt Berlin [u.a.] <<de>> Gruyter 2003 XXI, 364 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier De Gruyter studies in mathematics 29 Literaturverz. S. 355 - 357 Groupes discontinus Groupes discontinus ram Isométrie (Mathématiques) Isométrie (mathématiques) ram Discontinuous groups Isometrics (Mathematics) Isometriegruppe (DE-588)4162531-6 gnd rswk-swf Diskrete Gruppe (DE-588)4135541-6 gnd rswk-swf Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Hyperbolische Geometrie (DE-588)4161041-6 gnd rswk-swf Hyperbolische Geometrie (DE-588)4161041-6 s Riemannsche Fläche (DE-588)4049991-1 s Isometriegruppe (DE-588)4162531-6 s Diskrete Gruppe (DE-588)4135541-6 s DE-604 Nielsen, Jakob 1890-1959 Verfasser (DE-588)117730459 aut Schmidt, Asmus L. Sonstige oth De Gruyter studies in mathematics 29 (DE-604)BV000005407 29 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010170003&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fenchel, Werner 1905-1988 Nielsen, Jakob 1890-1959 Discontinuous groups of isometries in the hyperbolic plane De Gruyter studies in mathematics Groupes discontinus Groupes discontinus ram Isométrie (Mathématiques) Isométrie (mathématiques) ram Discontinuous groups Isometrics (Mathematics) Isometriegruppe (DE-588)4162531-6 gnd Diskrete Gruppe (DE-588)4135541-6 gnd Riemannsche Fläche (DE-588)4049991-1 gnd Hyperbolische Geometrie (DE-588)4161041-6 gnd |
| subject_GND | (DE-588)4162531-6 (DE-588)4135541-6 (DE-588)4049991-1 (DE-588)4161041-6 |
| title | Discontinuous groups of isometries in the hyperbolic plane |
| title_auth | Discontinuous groups of isometries in the hyperbolic plane |
| title_exact_search | Discontinuous groups of isometries in the hyperbolic plane |
| title_full | Discontinuous groups of isometries in the hyperbolic plane Werner Fenchel ; Jakob Nielsen. Ed. by Asmus L. Schmidt |
| title_fullStr | Discontinuous groups of isometries in the hyperbolic plane Werner Fenchel ; Jakob Nielsen. Ed. by Asmus L. Schmidt |
| title_full_unstemmed | Discontinuous groups of isometries in the hyperbolic plane Werner Fenchel ; Jakob Nielsen. Ed. by Asmus L. Schmidt |
| title_short | Discontinuous groups of isometries in the hyperbolic plane |
| title_sort | discontinuous groups of isometries in the hyperbolic plane |
| topic | Groupes discontinus Groupes discontinus ram Isométrie (Mathématiques) Isométrie (mathématiques) ram Discontinuous groups Isometrics (Mathematics) Isometriegruppe (DE-588)4162531-6 gnd Diskrete Gruppe (DE-588)4135541-6 gnd Riemannsche Fläche (DE-588)4049991-1 gnd Hyperbolische Geometrie (DE-588)4161041-6 gnd |
| topic_facet | Groupes discontinus Isométrie (Mathématiques) Isométrie (mathématiques) Discontinuous groups Isometrics (Mathematics) Isometriegruppe Diskrete Gruppe Riemannsche Fläche Hyperbolische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010170003&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005407 |
| work_keys_str_mv | AT fenchelwerner discontinuousgroupsofisometriesinthehyperbolicplane AT nielsenjakob discontinuousgroupsofisometriesinthehyperbolicplane AT schmidtasmusl discontinuousgroupsofisometriesinthehyperbolicplane |