Richard Thompson's famous group $F$ has the striking property that it can be realized as a dense subgroup of the group of all orientation-preserving homeomorphisms of the unit interval, but it can also be given by a simple 2-generator-2-relator presentation, in fact as the fundamental group of an aspherical complex with only two cells in each dimension.
This monograph studies a natural generalization of $F$ that also includes Melanie Stein's generalized $F$-groups. The main aims of this monograph are the determination of isomorphisms among the generalized $F$-groups and the study of their automorphism groups. This book is aimed at graduate students (or teachers of graduate students) interested in a class of examples of torsion-free infinite groups with elements and composition that are easy to describe and work with, but have unusual properties and surprisingly small presentations in terms of generators and defining relations.
| Preface |
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vii | |
| Introduction |
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1 | (12) |
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1 | (3) |
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2 Outline of our investigation |
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4 | (9) |
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Chapter A Construction of Finitary PL-homeomorphisms |
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13 | (10) |
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13 | (2) |
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15 | (2) |
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17 | (6) |
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Chapter B Generating Sets |
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23 | (20) |
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6 Necessary conditions for finite generation |
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23 | (1) |
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7 Generators and relations for groups with supports in the line |
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23 | (1) |
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8 Generators and relations for groups with supports in a half line |
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24 | (4) |
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9 Generators for groups with supports in a compact interval |
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28 | (15) |
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Chapter C The Subgroup of Bounded Homeomorphisms B |
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43 | (18) |
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10 Simplicity of the derived group of the subgroup B |
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43 | (2) |
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11 Construction of homomorphisms into the slope group |
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45 | (4) |
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12 Investigation of the abelianization of the subgroup B |
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49 | (12) |
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61 | (28) |
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13 Presentations of groups with supports in the line |
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61 | (10) |
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14 Presentations of groups with supports in a half line |
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71 | (14) |
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15 Presentations of groups with supports in a compact interval |
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85 | (4) |
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Chapter E Isomorphisms and Automorphism Groups |
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89 | (48) |
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89 | (12) |
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17 Isomorphisms of groups with non-cyclic slope groups |
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101 | (9) |
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18 Isomorphisms of groups with cyclic slope groups |
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110 | (15) |
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19 Automorphism groups of groups with cyclic slope groups |
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125 | (12) |
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137 | (26) |
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N1 Differences between memoir and monograph: summary |
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137 | (1) |
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N2 Differences between memoir and monograph: details |
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138 | (7) |
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145 | (18) |
| Bibliography |
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163 | (4) |
| Index of Notation |
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167 | (4) |
| Subject Index |
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171 | |
Robert Bieri, Johann Wolfgang Goethe-Universitat Frankfurt, Frankurt am Main, Germany.
Ralph Strebel, Universite de Fribourg, Switzerland.
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