A universal construction for groups acting freely on real trees

300 Pages·2012·1.021 MB·English

by  Chiswell, Ian, Müller, T. W

CAMBRIDGE TRACTS IN MATHEMATICS
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B. BOLLOBA´ S, W. FULTON, A. KATOK,
F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO
195 AUniversalConstructionforGroupsActingFreelyonRealTrees
CAMBRIDGE TRACTS IN MATHEMATICS
GENERAL EDITORS
B. BOLLOBA´S, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK,
B. SIMON, B.TOTARO
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A Universal Construction for Groups
Acting Freely on Real Trees
IAN CHISWELL
QueenMary,UniversityofLondon
THOMAS MU¨ LLER
QueenMary,UniversityofLondon
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(cid:2)C IanChiswellandThomasMu¨ller2012
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Contents
Preface pagexii
1 Introduction 1
1.1 Finitewordsandfreegroups 1
1.2 WordsoveradiscretelyorderedabeliangroupΛ 2
1.3 ThecasewhereΛisdenselyordered 4
1.4 ThecasewhereΛ=R 5
1.5 Contentsofthebook 7
2 ThegroupRF(G) 13
2.1 Themonoid(F(G),∗) 13
2.2 Reducedfunctionsandreducedmultiplication 17
2.3 CancellationtheoryforRF(G) 22
2.4 ProofofTheorem2.13 28
2.5 ThesubgroupG 31
0
2.6 AppendixtoChapter2 33
2.7 Exercises 34
viii Contents
3 TheR-treeX associatedwithRF(G) 35
G
3.1 Introduction 35
3.2 ConstructionofX 36
G
3.3 Completenessandtransitivity 38
3.4 Cyclicreduction 40
3.5 Classificationofelements 44
3.6 Boundedsubgroups 47
3.7 PresentingE(G) 51
3.8 Aremarkconcerninguniversality 55
3.9 ThedegreeofverticesofX 58
G
3.10 Exercises 59
4 FreeR-treeactionsanduniversality 61
4.1 Introduction 61
4.2 Anembeddingtheorem 62
4.3 UniversalityofRF-groupsandtheirassociatedR-trees 71
4.4 Exercises 76
5 Exponentsums 78
5.1 Introduction 78
5.2 Somemeasuretheory 79
5.3 Themapsμ 82
g
5.4 Themapse 84
g
5.5 Themape 86
G
6 Functoriality 90
6.1 Introduction 90
Contents ix
6.2 ThefunctorR(cid:2)F(−) 92
6.3 ThefunctorR(cid:3)F(−) 98
6.4 ThefunctorR(cid:2)F (−) 101
0
6.5 A remark concerning the automorphism group of
RF(G)/E(G) 108
6.6 Exercises 112
7 Conjugacyofhyperbolicelements 113
7.1 Introduction 113
7.2 Theequivalencerelationτ andtheconjugacytheorem 116
G
7.3 Normalisersofinfinitecyclichyperbolicsubgroups 118
7.4 Themainlemma 121
7.5 ProofofTheorem7.5 124
7.6 Exercises 124
8 Thecentralisersofhyperbolicelements 125
8.1 Introduction 125
8.2 Apreliminarylemma 126
8.3 Theperiodsofahyperbolicfunction 127
8.4 ThesubsetC−ofC 131
f f
8.5 ThesubsetC+ofC 135
f f
8.6 ThesubsetC ofC 139
f f
8.7 Themainresult 143
8.8 ThecasewhenC iscyclic 150
f
8.9 An application: the non-existence of soluble normal
subgroups 151