Two-generator discrete subgroups of PSL(2,R):
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
1995
|
| Series: | American Mathematical Society: Memoirs of the American Mathematical Society
561 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Volume 117, Number 561 (fourth of 5 numbers) |
| Physical Description: | X, 204 S. graph. Darst. |
| ISBN: | 0821803611 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Contents
I Introduction 1
1 Introduction 3
1.1 Overview Intersecting Axes 5
1.2 Overview of the intertwining cases 8
1.3 Why an algorithm is needed 10
2 The Acute Triangle Theorem 13
2.1 Nielsen equivalence 13
2.1.1 Groups and Triangles 14
2.1.2 The Geometric Description of Trace Minimizing . 15
2.1.3 Groups and pentagons 18
2.2 Idea of proof: Acute triangle theorem 19
2.3 Labeling Conventions 20
2.4 Ascending order conventions 21
2.5 The Triangle Algorithm 21
2.6 Q and the last triangle along A 25
2.7 Combining triangle algorithm steps 28
2.8 The sides and heights converge to 0 31
2.9 Acute triangle theorem: proof 31
3 Discreteness Theorem Proof Outline 33
3.1 The Discreteness Theorem 33
3.2 Discreteness theorem 35
3.3 Geometric equivalence theorems 36
II Preliminaries 39
V
vi CONTENTS
4 Triangle Groups and their Tilings 41
4.1 Basic facts about triangle groups 42
4.2 Minimal tiling distances 44
4.3 The wedge at a vertex 46
4.4 Proofs of lemmas and theorems 50
4.5 Additional Notation 55
4.6 Distances in the extended wedge 55
5 Pentagons 61
5.1 Constructing the pentagon, Pa,b 61
5.2 Notation 66
5.3 Applying the Poincare Polygon Theorem 66
5.4 Pentagon Tilings 67
5.5 Distances in the shingling 68
5.6 Distance lemmas in P 71
6 Hyperbolic Formulae Geometry 73
6.1 Addition formulas for the hyperbolic sine and cosine ... 73
6.2 Hyperbolic Triangles 74
6.3 Geometric Corollaries and Lemmas 75
6.3.1 Consequences of the Hyperbolic Law of Cosines . 75
6.3.2 Geometric corollaries for quadrilaterals and pen¬
tagons 76
6.3.3 The geometry of acute and equilateral triangles . 78
7 Extending Knapp Poincare 81
7.1 Extending the Knapp Count 82
7.2 Facts and Notation 83
7.3 Restrict to P = PA B 85
7.4 Counting the images of vertices 91
III Geometric Equivalence and the Discrete¬
ness Theorem 95
8 The Standard Acute Triangles 97
8.1 Stact: (2,3,n)t = 3, k = 3 98
CONTENTS vii
8.2 Stact: (2,4,n)i = 2; k = 2 102
8.3 Stact: (2,3,7)2 = 9; k = 2 103
8.4 Pentagon distances (2,3, n)t = 3; k = 3 107
9 Nielsen Eq: (2,3,n)* = S;k = 3 109
9.1 Introduction 109
9.2 Types of triples: distances 110
9.3 Locating t2 and t3 110
10 Nielsen Eq: (2,4,n)t = 2; jfc = 2 113
10.1 Introduction 113
10.2 Types of triples 114
10.2.1 Distances in the standard pentagon 115
10.3 Location of ti,t2 and t
11 Pentagon t = 9 2 2 Spectrum 121
11.1 Step 1: Label the wedge 122
11.2 Step 2: Double and Extend 123
11.3 Step 3: Drop perpendiculars 126
11.4 The two two spectrum 129
11.5 More distance computations 130
11.6 Distances to q o 132
11.7 Locate three order two points 140
12 The Seven Geometric Eq t = 9 141
12.1 Introduction 141
12.2 The variation of h and b 142
12.3 Rule out a seven on the j3 side 144
12.4 Rule out a seven on the D side 144
12.5 Interior sevens 146
12.6 Notation 148
12.6.1 b = T0 148
12.6.2 Rule out b = 2T0 = 7 149
12.6.3 Rule out {q, A ^qJ.q } 150
12.6.4 Rule out b = T2 151
12.7 Geometric equivalence (2,3, 7)/ = 9; k = 2 156
viii CONTENTS
13 Discreteness Theorem Proof 159
13.1 The Proof of the Discreteness Theorem 159
13.2 The proof of sufficiency 161
IV The Real Number Algorithm and the Tur¬
ing Machine Algorithm 165
14 Forms of the Algorithm 167
14.1 What is an algorithm? 168
14.1.1 Turing machine Algorithms 170
14.1.2 Real Number Algorithms 171
14.2 The Elliptic Order Algorithm 173
14.3 Notation and Preliminary Computations 176
14.3.1 Fixed points and cross ratios 176
14.3.2 Computation of P2 and 7 177
14.3.3 Positive and Negative Rotations 178
14.4 The Real Number Algorithm 179
14.5 The Turing Machine Algorithm 190
V Appendix 193
A Verify Matelski Beardon Count 195
B A Summary of Notation 199
|
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| classification_rvk | SI 130 |
| classification_tum | MAT 203f |
| ctrlnum | (OCoLC)246283999 (DE-599)BVBBV010463930 |
| discipline | Mathematik |
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| id | DE-604.BV010463930 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:52:56Z |
| institution | BVB |
| isbn | 0821803611 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006972265 |
| oclc_num | 246283999 |
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| physical | X, 204 S. graph. Darst. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | American Mathematical Society: Memoirs of the American Mathematical Society |
| series2 | American Mathematical Society: Memoirs of the American Mathematical Society |
| spelling | Gilman, Jane P. 1945- Verfasser (DE-588)134012801 aut Two-generator discrete subgroups of PSL(2,R) Jane Gilman Providence, RI American Math. Soc. 1995 X, 204 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier American Mathematical Society: Memoirs of the American Mathematical Society 561 Volume 117, Number 561 (fourth of 5 numbers) Diskrete Untergruppe (DE-588)4257236-8 gnd rswk-swf Unimodulare Gruppe (DE-588)4186897-3 gnd rswk-swf Unimodulare Gruppe (DE-588)4186897-3 s Diskrete Untergruppe (DE-588)4257236-8 s DE-604 American Mathematical Society: Memoirs of the American Mathematical Society 561 (DE-604)BV008000141 561 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006972265&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gilman, Jane P. 1945- Two-generator discrete subgroups of PSL(2,R) American Mathematical Society: Memoirs of the American Mathematical Society Diskrete Untergruppe (DE-588)4257236-8 gnd Unimodulare Gruppe (DE-588)4186897-3 gnd |
| subject_GND | (DE-588)4257236-8 (DE-588)4186897-3 |
| title | Two-generator discrete subgroups of PSL(2,R) |
| title_auth | Two-generator discrete subgroups of PSL(2,R) |
| title_exact_search | Two-generator discrete subgroups of PSL(2,R) |
| title_full | Two-generator discrete subgroups of PSL(2,R) Jane Gilman |
| title_fullStr | Two-generator discrete subgroups of PSL(2,R) Jane Gilman |
| title_full_unstemmed | Two-generator discrete subgroups of PSL(2,R) Jane Gilman |
| title_short | Two-generator discrete subgroups of PSL(2,R) |
| title_sort | two generator discrete subgroups of psl 2 r |
| topic | Diskrete Untergruppe (DE-588)4257236-8 gnd Unimodulare Gruppe (DE-588)4186897-3 gnd |
| topic_facet | Diskrete Untergruppe Unimodulare Gruppe |
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