How groups grow:
Growth of groups is an innovative new branch of group theory. This is the first book to introduce the subject from scratch. It begins with basic definitions and culminates in the seminal results of Gromov and Grigorchuk and more. The proof of Gromov's theorem on groups of polynomial growth is g...
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2012
|
| Series: | London Mathematical Society lecture note series
395 |
| Subjects: | |
| Online Access: | BSB01 FHN01 Volltext |
| Summary: | Growth of groups is an innovative new branch of group theory. This is the first book to introduce the subject from scratch. It begins with basic definitions and culminates in the seminal results of Gromov and Grigorchuk and more. The proof of Gromov's theorem on groups of polynomial growth is given in full, with the theory of asymptotic cones developed on the way. Grigorchuk's first and general groups are described, as well as the proof that they have intermediate growth, with explicit bounds, and their relationship to automorphisms of regular trees and finite automata. Also discussed are generating functions, groups of polynomial growth of low degrees, infinitely generated groups of local polynomial growth, the relation of intermediate growth to amenability and residual finiteness, and conjugacy class growth. This book is valuable reading for researchers, from graduate students onward, working in contemporary group theory |
| Item Description: | Title from publisher's bibliographic system (viewed on 05 Oct 2015) |
| Physical Description: | 1 online resource (ix, 199 pages) |
| ISBN: | 9781139095129 |
| DOI: | 10.1017/CBO9781139095129 |
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| 505 | 8 | 0 | |g 1 |g 1 |t Introduction |g 2 |g 15 |t Some Group Theory |g 2.1 |g 15 |t Finite Index Subgroups |g 2.2 |g 18 |t Growth |g 2.3 |g 25 |t Soluble and Polycyclic Groups |g 2.4 |g 27 |t Nilpotent Groups |g 2.5 |g 32 |t Isoperimetric Inequalities |g 3 |g 36 |t Groups of Linear Growth |g 3.1 |g 36 |t Linear Growth |g 3.2 |g 41 |t Linear Growth Functions |g 4 |g 44 |t The Growth of Nilpotent Groups |g 4.1 |g 44 |t Polynomial Growth of Nilpotent Groups |g 4.2 |g 50 |t Groups of Small Degree |g 5 |g 56 |t The Growth of Soluble Groups |g 5.1 |g 56 |t Soluble Groups of Polynomial Growth |g 5.2 |g 60 |t Uniform Exponential Growth of Soluble Groups |g 6 |g 63 |t Linear Groups |g 7 |g 67 |t Asymptotic Cones |g 8 |g 77 |t Groups of Polynomial Growth |g 9 |g 81 |t Infinitely Generated Groups |g 10 |g 90 |t Intermediate Growth: Grigorchuk's First Group |g 11 |g 108 |t More Groups of Intermediate Growth |g 11.1 |g 108 |t The General Grigorchuk Groups |g 11.2 |g 113 |t Groups Acting on Regular Trees |g 11.3 |g 115 |t Groups Defined by Finite Automata |g 11.4 |g 119 |t Bartholdi-Erschler Groups |g 12 |g 121 |t Growth and Amenability |g 12.1 |g 121 |t Amenability and Intermediate Growth |g 12.2tMore Isoperimetric Inequalities |g 127 |g 13 |g 131 |t Intermediate Growth and Residual Finiteness |g 14 |g 136 |t Explicit Calculations |g 14.1 |g 136 |t The Trefoil Group |g 14.2 |g 139 |t Wreath Products |
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| author | Mann, Avinoam 1937- |
| author_facet | Mann, Avinoam 1937- |
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| author_sort | Mann, Avinoam 1937- |
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| contents | Introduction Some Group Theory Finite Index Subgroups Growth Soluble and Polycyclic Groups Nilpotent Groups Isoperimetric Inequalities Groups of Linear Growth Linear Growth Linear Growth Functions The Growth of Nilpotent Groups Polynomial Growth of Nilpotent Groups Groups of Small Degree The Growth of Soluble Groups Soluble Groups of Polynomial Growth Uniform Exponential Growth of Soluble Groups Linear Groups Asymptotic Cones Groups of Polynomial Growth Infinitely Generated Groups Intermediate Growth: Grigorchuk's First Group More Groups of Intermediate Growth The General Grigorchuk Groups Groups Acting on Regular Trees Groups Defined by Finite Automata Bartholdi-Erschler Groups Growth and Amenability Amenability and Intermediate Growth Intermediate Growth and Residual Finiteness Explicit Calculations The Trefoil Group Wreath Products |
| ctrlnum | (ZDB-20-CBO)CR9781139095129 (OCoLC)852522294 (DE-599)BVBBV043941857 |
| dewey-full | 512.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.2 |
| dewey-search | 512.2 |
| dewey-sort | 3512.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1017/CBO9781139095129 |
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| id | DE-604.BV043941857 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:39:16Z |
| institution | BVB |
| isbn | 9781139095129 |
| language | English |
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| physical | 1 online resource (ix, 199 pages) |
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| publishDate | 2012 |
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| spelling | Mann, Avinoam 1937- Verfasser aut How groups grow Avinoam Mann Cambridge Cambridge University Press 2012 1 online resource (ix, 199 pages) txt rdacontent c rdamedia cr rdacarrier London Mathematical Society lecture note series 395 Title from publisher's bibliographic system (viewed on 05 Oct 2015) 1 1 Introduction 2 15 Some Group Theory 2.1 15 Finite Index Subgroups 2.2 18 Growth 2.3 25 Soluble and Polycyclic Groups 2.4 27 Nilpotent Groups 2.5 32 Isoperimetric Inequalities 3 36 Groups of Linear Growth 3.1 36 Linear Growth 3.2 41 Linear Growth Functions 4 44 The Growth of Nilpotent Groups 4.1 44 Polynomial Growth of Nilpotent Groups 4.2 50 Groups of Small Degree 5 56 The Growth of Soluble Groups 5.1 56 Soluble Groups of Polynomial Growth 5.2 60 Uniform Exponential Growth of Soluble Groups 6 63 Linear Groups 7 67 Asymptotic Cones 8 77 Groups of Polynomial Growth 9 81 Infinitely Generated Groups 10 90 Intermediate Growth: Grigorchuk's First Group 11 108 More Groups of Intermediate Growth 11.1 108 The General Grigorchuk Groups 11.2 113 Groups Acting on Regular Trees 11.3 115 Groups Defined by Finite Automata 11.4 119 Bartholdi-Erschler Groups 12 121 Growth and Amenability 12.1 121 Amenability and Intermediate Growth 12.2tMore Isoperimetric Inequalities 127 13 131 Intermediate Growth and Residual Finiteness 14 136 Explicit Calculations 14.1 136 The Trefoil Group 14.2 139 Wreath Products Growth of groups is an innovative new branch of group theory. This is the first book to introduce the subject from scratch. It begins with basic definitions and culminates in the seminal results of Gromov and Grigorchuk and more. The proof of Gromov's theorem on groups of polynomial growth is given in full, with the theory of asymptotic cones developed on the way. Grigorchuk's first and general groups are described, as well as the proof that they have intermediate growth, with explicit bounds, and their relationship to automorphisms of regular trees and finite automata. Also discussed are generating functions, groups of polynomial growth of low degrees, infinitely generated groups of local polynomial growth, the relation of intermediate growth to amenability and residual finiteness, and conjugacy class growth. This book is valuable reading for researchers, from graduate students onward, working in contemporary group theory Group theory Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Wachstum (DE-588)4064115-6 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 s Wachstum (DE-588)4064115-6 s 1\p DE-604 Erscheint auch als Druckausgabe 978-1-107-65750-2 https://doi.org/10.1017/CBO9781139095129 Verlag URL des Erstveröffentlichers Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Mann, Avinoam 1937- How groups grow Introduction Some Group Theory Finite Index Subgroups Growth Soluble and Polycyclic Groups Nilpotent Groups Isoperimetric Inequalities Groups of Linear Growth Linear Growth Linear Growth Functions The Growth of Nilpotent Groups Polynomial Growth of Nilpotent Groups Groups of Small Degree The Growth of Soluble Groups Soluble Groups of Polynomial Growth Uniform Exponential Growth of Soluble Groups Linear Groups Asymptotic Cones Groups of Polynomial Growth Infinitely Generated Groups Intermediate Growth: Grigorchuk's First Group More Groups of Intermediate Growth The General Grigorchuk Groups Groups Acting on Regular Trees Groups Defined by Finite Automata Bartholdi-Erschler Groups Growth and Amenability Amenability and Intermediate Growth Intermediate Growth and Residual Finiteness Explicit Calculations The Trefoil Group Wreath Products Group theory Gruppentheorie (DE-588)4072157-7 gnd Wachstum (DE-588)4064115-6 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4064115-6 |
| title | How groups grow |
| title_alt | Introduction Some Group Theory Finite Index Subgroups Growth Soluble and Polycyclic Groups Nilpotent Groups Isoperimetric Inequalities Groups of Linear Growth Linear Growth Linear Growth Functions The Growth of Nilpotent Groups Polynomial Growth of Nilpotent Groups Groups of Small Degree The Growth of Soluble Groups Soluble Groups of Polynomial Growth Uniform Exponential Growth of Soluble Groups Linear Groups Asymptotic Cones Groups of Polynomial Growth Infinitely Generated Groups Intermediate Growth: Grigorchuk's First Group More Groups of Intermediate Growth The General Grigorchuk Groups Groups Acting on Regular Trees Groups Defined by Finite Automata Bartholdi-Erschler Groups Growth and Amenability Amenability and Intermediate Growth Intermediate Growth and Residual Finiteness Explicit Calculations The Trefoil Group Wreath Products |
| title_auth | How groups grow |
| title_exact_search | How groups grow |
| title_full | How groups grow Avinoam Mann |
| title_fullStr | How groups grow Avinoam Mann |
| title_full_unstemmed | How groups grow Avinoam Mann |
| title_short | How groups grow |
| title_sort | how groups grow |
| topic | Group theory Gruppentheorie (DE-588)4072157-7 gnd Wachstum (DE-588)4064115-6 gnd |
| topic_facet | Group theory Gruppentheorie Wachstum |
| url | https://doi.org/10.1017/CBO9781139095129 |
| work_keys_str_mv | AT mannavinoam howgroupsgrow |