Topological Groups

Topological groupsFrom Wikipedia, the free encyclopedia

Contents

1 Adelic algebraic group 11.1 Ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Tamagawa numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 History of the terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Almost periodic function 32.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Uniform or Bohr or Bochner almost periodic functions . . . . . . . . . . . . . . . . . . . . 32.1.2 Stepanov almost periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.3 Weyl almost periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.4 Besicovitch almost periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.5 Almost periodic functions on a locally compact abelian group . . . . . . . . . . . . . . . . 5

2.2 Quasiperiodic signals in audio and music synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Amenable group 93.1 Denition for locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Equivalent conditions for amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Case of discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Basic subgroup 15

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4.1 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Generalization to modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Bohr compactication 165.1 Denitions and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Maximally almost periodic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Cantor cube 186.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 Chabauty topology 197.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 Circle group 208.1 Elementary introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.2 Topological and analytic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.6 Group structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

9 Commensurability (mathematics) 259.1 History of the concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.2 Commensurability in group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.3 In topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.4 In physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

10 Commensurator 2710.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11 Compact group 28

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11.1 Compact Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.1.1 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

11.2 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.3 Haar measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.4 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.6 From compact to non-compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

12 Compactly generated group 3212.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.2 Locally compact case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

13 Continuous group action 3313.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

14 Covering group 3414.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.2 Group structure on a covering space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414.3 Universal covering group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.4 Lattice of covering groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.5 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

15 Descendant tree (group theory) 3715.1 Denitions and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715.2 Pro-p groups and coclass trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.3 Tree diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.4 Virtual periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3815.5 Multifurcation and coclass graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015.6 Identiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015.7 Concrete examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

15.7.1 Coclass 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015.7.2 Coclass 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4115.7.3 Coclass 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.7.4 Coclass 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

15.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4715.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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15.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

16 Direct sum of topological groups 5016.1 Topological direct summands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5016.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5016.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

17 Elementary amenable group 5217.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

18 Extension of a topological group 5318.1 Clasication of extensions of topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.3 Extensions of locally compact abelian groups (LCA) . . . . . . . . . . . . . . . . . . . . . . . . . 5418.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

19 FC-group 5519.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5519.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

20 Free regular set 5620.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5620.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5620.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

21 Free-by-cyclic group 5721.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

22 Fundamental domain 5822.1 Hints at general denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5822.3 Fundamental domain for the modular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6022.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6022.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

23 Gromovs theorem on groups of polynomial growth 6223.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

24 Growth rate (group theory) 6424.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6424.2 Polynomial and exponential growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6524.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6524.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6624.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6624.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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25 Haar measure 6725.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6725.2 Haars theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6725.3 Construction of Haar measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

25.3.1 A construction using compact subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6825.3.2 A construction using compactly supported functions . . . . . . . . . . . . . . . . . . . . . 6925.3.3 A construction using mean values of functions . . . . . . . . . . . . . . . . . . . . . . . . 6925.3.4 A construction on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

25.4 The right Haar measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6925.4.1 The modular function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

25.5 Measures on homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7025.6 Haar integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7025.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7125.8 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

25.8.1 Abstract harmonic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7225.8.2 Mathematical statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

25.9 Weils converse theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7225.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7225.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7225.12Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7325.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

26 Halls universal group 7426.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7426.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

27 Ulms theorem 7527.1 Denition of height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7527.2 Ulm subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7527.3 Ulms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

27.3.1 Alternative formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7627.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

28 Higmans embedding theorem 7828.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

29 HilbertSmith conjecture 7929.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

30 HirschPlotkin radical 8030.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

31 Homeomorphism group 8131.1 Properties and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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31.1.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8131.2 Mapping class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8131.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8231.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

32 Homogeneous space 8332.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

32.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8432.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8432.3 Homogeneous spaces as coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8532.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8532.5 Prehomogeneous vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8532.6 Homogeneous spaces in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8632.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8632.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

33 Hopan group 8733.1 Examples of Hopan groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8733.2 Examples of non-Hopan groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8733.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8733.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8833.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

34 Identity component 8934.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8934.2 Component group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8934.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8934.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

35 Innite conjugacy class property 9135.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

36 Innite dihedral group 9236.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9236.2 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9236.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

37 Innite group 9437.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

38 Kazhdans property (T) 9538.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9538.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9538.3 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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38.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9638.5 Discrete groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9638.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9738.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

39 Kroneckers theorem 9939.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9939.2 Relation to tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9939.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10039.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

40 Locally compact group 10140.1 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10140.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10140.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10240.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

41 Locally nite group 10341.1 Denition and rst consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10341.2 Examples and non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10341.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10441.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10441.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

42 Locally pronite group 10542.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10542.2 Representations of a locally pronite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10542.3 Hecke algebra of a locally pronite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10642.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10642.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

43 Loop group 10743.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10743.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10743.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10843.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10843.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

44 Matrix group 10944.1 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10944.2 Classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10944.3 Finite groups as matrix groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10944.4 Representation theory and character theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11044.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

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44.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11044.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11044.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

45 Mautners lemma 11145.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

46 Maximal compact subgroup 11246.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11246.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11246.3 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

46.3.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11346.3.2 Proof of uniqueness for semisimple groups . . . . . . . . . . . . . . . . . . . . . . . . . . 113

46.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11446.4.1 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11446.4.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

46.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11446.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11546.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

47 Monothetic group 11647.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

48 No small subgroup 11748.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

49 Noncommutative harmonic analysis 11849.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11849.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11849.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

50 Oligomorphic group 12050.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12050.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

51 One-parameter group 12151.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12151.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12251.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12251.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12251.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

52 Paratopological group 12352.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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53 PeterWeyl theorem 12453.1 Matrix coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12453.2 Decomposition of a unitary representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12553.3 Decomposition of square-integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12553.4 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

53.4.1 Structure of compact topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 12653.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12653.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

54 Pontryagin duality 12754.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12854.2 Locally compact abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

54.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12854.2.2 The dual group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

54.3 The Pontryagin duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12954.4 Pontryagin duality and the Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

54.4.1 Haar measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13054.4.2 Fourier transform and Fourier inversion formula for L1-functions . . . . . . . . . . . . . . 13054.4.3 The group algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13154.4.4 Plancherel and L2 Fourier inversion theorems . . . . . . . . . . . . . . . . . . . . . . . . 132

54.5 Bohr compactication and almost-periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13254.6 Categorical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13354.7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

54.7.1 Non-commutative theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13354.7.2 Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

54.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13454.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

55 Positive real numbers 13555.1 Logarithmic measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13555.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

56 Principal homogeneous space 13656.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13756.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13756.3 Other usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13856.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13856.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13856.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13856.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

57 Pro-p group 14057.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

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58 Pronite group 14158.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14158.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14158.3 Properties and facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14258.4 Pronite completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14258.5 Ind-nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14258.6 Projective pronite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14258.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14358.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

59 Properly discontinuous action 14459.1 Properly discontinuous action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

59.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14459.2 Similar properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

59.2.1 Wandering actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14459.2.2 Discrete orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145


60 Prosolvable group 14660.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14660.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14660.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

61 Protorus 14761.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14761.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

62 Prfer group 14862.1 Constructions of Z(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14862.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14962.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15062.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15062.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

63 Prfer rank 15163.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15163.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15163.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

64 Prfer theorems 152

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64.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15264.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

65 Quasiregular representation 153

66 Residual property (mathematics) 15466.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15466.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

67 Residually nite group 15567.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15567.2 Pronite topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15567.3 Varieties of residually nite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15667.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15667.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

68 Residue-class-wise ane group 15768.1 References and external links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

69 Restricted product 15969.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15969.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

70 SchwartzBruhat function 16070.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16070.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16070.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

71 Solenoid (mathematics) 16171.1 Geometric construction and the SmaleWilliams attractor . . . . . . . . . . . . . . . . . . . . . . 16271.2 Pathological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16371.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16371.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16371.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

72 Stable group 16572.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16572.2 The CherlinZilber conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16572.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

73 Subgroup growth 16773.1 Nilpotent groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16773.2 Congruence subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16973.3 Subgroup growth and coset representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16973.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

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74 System of imprimitivity 17074.1 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

74.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17174.2 Innite dimensional systems of imprimitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

74.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17274.3 Homogeneous systems of imprimitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

74.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17474.4 Induced representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17474.5 Applications to the theory of group representations . . . . . . . . . . . . . . . . . . . . . . . . . 174

74.5.1 Example: the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17574.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

75 Tame group 17875.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

76 TannakaKrein duality 17976.1 The idea of TannakaKrein duality: category of representations of a group . . . . . . . . . . . . . 17976.2 Theorems of Tannaka and Krein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17976.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18076.4 DoplicherRoberts theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18076.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18076.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

77 Tarski monster group 18277.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18277.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18277.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

78 Thompson groups 18378.1 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18378.2 Other representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18378.3 Amenability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18478.4 Connections with topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18478.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18578.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

79 Tits alternative 18679.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18679.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18679.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

80 Topological abelian group 18780.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18780.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

CONTENTS xiii

81 Topological group 18881.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

81.1.1 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18981.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18981.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18981.4 Relationship to other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19081.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19081.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19081.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19181.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

82 Topological semigroup 19282.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19282.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

83 Totally disconnected group 19383.1 Locally compact case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

83.1.1 Tidy subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19383.1.2 The scale function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19483.1.3 Calculations and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

83.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19483.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

84 Ulms theorem 19584.1 Denition of height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19584.2 Ulm subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19584.3 Ulms theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

84.3.1 Alternative formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19684.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

85 Verbal subgroup 19885.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

86 Von Neumann conjecture 19986.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

87 Z-group 20187.1 Groups whose Sylow subgroups are cyclic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20187.2 Group with a generalized central series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20187.3 Special 2-transitive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20287.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20287.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 203

87.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20387.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

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87.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Chapter 1

Adelic algebraic group

In abstract algebra, an adelic algebraic group is a semitopological group dened by an algebraic group G over anumber eld K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the denition ofthe appropriate topology is straightforward only in caseG is a linear algebraic group. In the case ofG an abelian varietyit presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawanumbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphicrepresentations, and the arithmetic of quadratic forms.In case G is a linear algebraic group, it is an ane algebraic variety in ane N-space. The topology on the adelicalgebraic groupG(A) is taken to be the subspace topology in AN , the Cartesian product of N copies of the adele ring.

1.1 IdelesAn important example, the idele group I(K), is the case of G = GL1 . Here the set of ideles (also idles /dlz/)consists of the invertible adeles; but the topology on the idele group is not their topology as a subset of the adeles.Instead, considering that GL1 lies in two-dimensional ane space as the 'hyperbola' dened parametrically by

{(t, t1)},

the topology correctly assigned to the idele group is that induced by inclusion in A2; composing with a projection, itfollows that the ideles carry a ner topology than the subspace topology from A.Inside AN , the product KN lies as a discrete subgroup. This means that G(K) is a discrete subgroup of G(A), also. Inthe case of the idele group, the quotient group

I(K)/K

is the idele class group. It is closely related to (though larger than) the ideal class group. The idele class group is notitself compact; the ideles must rst be replaced by the ideles of norm 1, and then the image of those in the idele classgroup is a compact group; the proof of this is essentially equivalent to the niteness of the class number.The study of the Galois cohomology of idele class groups is a central matter in class eld theory. Characters of theidele class group, now usually called Hecke characters, give rise to the most basic class of L-functions.

1.2 Tamagawa numbersSee also: Weil conjecture on Tamagawa numbers

For more general G, the Tamagawa number is dened (or indirectly computed) as the measure of

G(A)/G(K).

1

2 CHAPTER 1. ADELIC ALGEBRAIC GROUP

Tsuneo Tamagawa's observation was that, starting from an invariant dierential form on G, dened over K, themeasure involved was well-dened: while could be replaced by c with c a non-zero element of K, the productformula for valuations in K is reected by the independence from c of the measure of the quotient, for the productmeasure constructed from on each eective factor. The computation of Tamagawa numbers for semisimple groupscontains important parts of classical quadratic form theory.

1.3 History of the terminologyHistorically the idles were introduced by Chevalley (1936) under the name "lment idal, which is ideal elementin French, which Chevalley (1940) then abbreviated to idle following a suggestion of Hasse. (In these papers healso gave the ideles a non-Hausdor topology.) This was to formulate class eld theory for innite extensions in termsof topological groups. Weil (1938) dened (but did not name) the ring of adeles in the function eld case and pointedout that Chevalleys group of Idealelemente was the group of invertible elements of this ring. Tate (1950) dened thering of adeles as a restricted direct product, though he called its elements valuation vectors rather than adeles.Chevalley (1951) dened the ring of adeles in the function eld case, under the name repartitions. The term adle(short for additive idles, and also a French womans name) was in use shortly afterwards (Jaard 1953) and mayhave been introduced by Andr Weil. The general construction of adelic algebraic groups by Ono (1957) followedthe algebraic group theory founded by Armand Borel and Harish-Chandra.

1.4 References Chevalley, Claude (1936), Gnralisation de la thorie du corps de classes pour les extensions innies.,

Journal de Mathmatiques Pures et Appliques (in French) 15: 359371, JFM 62.1153.02 Chevalley, Claude (1940), La thorie du corps de classes, Annals of Mathematics. Second Series 41: 394418, ISSN 0003-486X, JSTOR 1969013, MR 0002357

Chevalley, Claude (1951), Introduction to the Theory of Algebraic Functions of One Variable, MathematicalSurveys, No. VI, Providence, R.I.: American Mathematical Society, MR 0042164

Jaard, Paul (1953), Anneaux d'adles (d'aprs Iwasawa), Sminaire Bourbaki, Secrtariat mathmatique,Paris, MR 0157859

Ono, Takashi (1957), Sur une proprit arithmtique des groupes algbriques commutatifs, Bulletin de laSocit Mathmatique de France 85: 307323, ISSN 0037-9484, MR 0094362

Tate, John T. (1950), Fourier analysis in number elds, and Heckes zeta-functions, Algebraic Number The-ory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305347, ISBN 978-0-9502734-2-6, MR 0217026

Weil, Andr (1938), Zur algebraischen Theorie der algebraischen Funktionen., Journal fr Reine und Ange-wandte Mathematik (in German) 179: 129133, doi:10.1515/crll.1938.179.129, ISSN 0075-4102

1.5 External links Rapinchuk, A.S. (2001), Tamagawa number, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

Chapter 2

Almost periodic function

In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic towithin any desired level of accuracy, given suitably long, well-distributed almost-periods. The concept was rststudied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abram SamoilovitchBesicovitch, amongst others. There is also a notion of almost periodic functions on locally compact abelian groups,rst studied by John von Neumann.Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, butnot exactly. An example would be a planetary system, with planets in orbits moving with periods that are notcommensurable (i.e., with a period vector that is not proportional to a vector of integers). A theorem of Kroneckerfrom diophantine approximation can be used to show that any particular conguration that occurs once, will recur towithin any specied accuracy: if we wait long enough we can observe the planets all return to within a second of arcto the positions they once were in.

2.1 MotivationThere are several inequivalent denitions of almost periodic functions. The rst was given by Harald Bohr. Hisinterest was initially in nite Dirichlet series. In fact by truncating the series for the Riemann zeta function (s) tomake it nite, one gets nite sums of terms of the type

e(+it) logn

with s written as ( + it) the sum of its real part and imaginary part it. Fixing , so restricting attention to a singlevertical line in the complex plane, we can see this also as

ne(logn)it:

Taking a nite sum of such terms avoids diculties of analytic continuation to the region < 1. Here the 'frequencieslog n will not all be commensurable (they are as linearly independent over the rational numbers as the integers n aremultiplicatively independent which comes down to their prime factorizations).With this initial motivation to consider types of trigonometric polynomial with independent frequencies, mathematicalanalysis was applied to discuss the closure of this set of basic functions, in various norms.The theory was developed using other norms by Besicovitch, Stepanov, Weyl, von Neumann, Turing, Bochner andothers in the 1920s and 1930s.

2.1.1 Uniform or Bohr or Bochner almost periodic functionsBohr (1925) dened the uniformly almost-periodic functions as the closure of the trigonometric polynomials withrespect to the uniform norm

3

4 CHAPTER 2. ALMOST PERIODIC FUNCTION

jjf jj1 = supxjf(x)j

(on bounded functions f on R). In other words, a function f is uniformly almost periodic if for every > 0 there is anite linear combination of sine and cosine waves that is of distance less than from f with respect to the uniformnorm. Bohr proved that this denition was equivalent to the existence of a relatively dense set of almost-periods,for all > 0: that is, translations T() = T of the variable t making

jf(t+ T ) f(t)j < ":An alternative denition due to Bochner (1926) is equivalent to that of Bohr and is relatively simple to state:

A function f is almost periodic if every sequence {(t + Tn)} of translations of f has a subsequencethat converges uniformly for t in (, +).

The Bohr almost periodic functions are essentially the same as continuous functions on the Bohr compactication ofthe reals.

2.1.2 Stepanov almost periodic functionsThe space Sp of Stepanov almost periodic functions (for p 1) was introduced by V.V. Stepanov (1925). It containsthe space of Bohr almost periodic functions. It is the closure of the trigonometric polynomials under the norm

jjf jjS;r;p = supx

1

r

Z x+rx

jf(s)jp ds1/p

for any xed positive value of r; for dierent values of r these norms give the same topology and so the same spaceof almost periodic functions (though the norm on this space depends on the choice of r).

2.1.3 Weyl almost periodic functionsThe spaceWp of Weyl almost periodic functions (for p 1) was introduced by Weyl (1927). It contains the space Spof Stepanov almost periodic functions. It is the closure of the trigonometric polynomials under the seminorm

jjf jjW;p = limr 7!1 jjf jjS;r;p

Warning: there are nonzero functions with ||||W, = 0, such as any bounded function of compact support, so to geta Banach space one has to quotient out by these functions.

2.1.4 Besicovitch almost periodic functionsThe space Bp of Besicovitch almost periodic functions was introduced by Besicovitch (1926). It is the closure of thetrigonometric polynomials under the seminorm

jjf jjB;p = lim supx!1

1

2x

Z xxjf(s)jp ds

1/pWarning: there are nonzero functions with ||||B,p = 0, such as any bounded function of compact support, so to geta Banach space one has to quotient out by these functions.The Besicovitch almost periodic functions in B2 have an expansion (not necessarily convergent) as

2.2. QUASIPERIODIC SIGNALS IN AUDIO AND MUSIC SYNTHESIS 5

Xane

int

with an2 nite and n real. Conversely every such series is the expansion of some Besicovitch periodic function(which is not unique).The space Bp of Besicovitch almost periodic functions (for p 1) contains the space Wp of Weyl almost periodicfunctions. If one quotients out a subspace of null functions, it can be identied with the space of Lp functions onthe Bohr compactication of the reals.

2.1.5 Almost periodic functions on a locally compact abelian groupWith these theoretical developments and the advent of abstract methods (the PeterWeyl theorem, Pontryagin dualityand Banach algebras) a general theory became possible. The general idea of almost-periodicity in relation to a locallycompact abelian group G becomes that of a function F in L(G), such that its translates by G form a relativelycompact set. Equivalently, the space of almost periodic functions is the norm closure of the nite linear combinationsof characters of G. If G is compact the almost periodic functions are the same as the continuous functions.The Bohr compactication ofG is the compact abelian group of all possibly discontinuous characters of the dual groupof G, and is a compact group containing G as a dense subgroup. The space of uniform almost periodic functions onG can be identied with the space of all continuous functions on the Bohr compactication of G. More generally theBohr compactication can be dened for any topological groupG, and the spaces of continuous or Lp functions on theBohr compactication can be considered as almost periodic functions on G. For locally compact connected groups Gthe map from G to its Bohr compactication is injective if and only if G is a central extension of a compact group,or equivalently the product of a compact group and a nite-dimensional vector space.

2.2 Quasiperiodic signals in audio and music synthesisIn speech processing, audio signal processing, and music synthesis, a quasiperiodic signal, sometimes called a quasi-harmonic signal, is a waveform that is virtually periodic microscopically, but not necessarily periodic macroscopi-cally. This does not give a quasiperiodic function in the sense of the Wikipedia article of that name, but somethingmore akin to an almost periodic function, being a nearly periodic function where any one period is virtually identicalto its adjacent periods but not necessarily similar to periods much farther away in time. This is the case for musi-cal tones (after the initial attack transient) where all partials or overtones are harmonic (that is all overtones are atfrequencies that are an integer multiple of a fundamental frequency of the tone).When a signal x(t) is fully periodic with period T , then the signal exactly satises

x(t) = x(t+ T )

or

jx(t) x(t+ T )j = 0 all for t:The Fourier series representation would be

x(t) =1

2a0 +

1Xn=1

[an cos(2nf0t) bn sin(2nf0t)]

or

x(t) =1

2a0 +

1Xn=1

[rn cos(2nf0t+ 'n)]


where f0 = 1T is the fundamental frequency and the Fourier coecients are

an = rn cos ('n) =2

T

Z t0+Tt0

x(t) cos(2nf0t) dt

bn = rn sin ('n) = 2T

Z t0+Tt0

x(t) sin(2nf0t) dt

where t0 can be any time: 1 < t0 < +1 .

The fundamental frequency f0 , and Fourier coecients an , bn , rn , or 'n , are constants, i.e. they are notfunctions of time. The harmonic frequencies are exact integer multiples of the fundamental frequency.When x(t) is quasiperiodic then

x(t) x (t+ T (t))or

jx(t) x (t+ T (t))j < "where

0 < kxk =px2 =

slim!1

1

Z /2/2

x2(t) dt:

Now the Fourier series representation would be

x(t) =1

2a0(t) +

1Xn=1

an(t) cos

2n

Z t0

f0() d

bn(t) sin

2n

Z t0

f0() d

or

x(t) =1

2a0(t) +

1Xn=1

rn(t) cos

2n

Z t0

f0() d + 'n(t)

or

x(t) =1

2a0(t) +

1Xn=1

rn(t) cos

2

Z t0

fn() d + 'n(0)

where f0(t) = 1T (t) is the possibly time-varying fundamental frequency and the Fourier coecients are

an(t) = rn(t) cos ('n(t))

bn(t) = rn(t) sin ('n(t))and the instantaneous frequency for each partial is

fn(t) = nf0(t) +1

2'0n(t):

2.3. SEE ALSO 7

Whereas in this quasiperiodic case, the fundamental frequency f0(t) , the harmonic frequencies fn(t) , and theFourier coecients an(t) , bn(t) , rn(t) , or 'n(t) are not necessarily constant, and are functions of time albeitslowly varying functions of time. Stated dierently these functions of time are bandlimited to much less than thefundamental frequency for x(t) to be considered to be quasiperiodic.The partial frequencies fn(t) are very nearly harmonic but not necessarily exactly so. The time-derivative of 'n(t), that is '0n(t) , has the eect of detuning the partials from their exact integer harmonic value nf0(t) . A rapidlychanging 'n(t) means that the instantaneous frequency for that partial is severely detuned from the integer harmonicvalue which would mean that x(t) is not quasiperiodic.

2.3 See also Quasiperiodic function Aperiodic function Quasiperiodic tiling Fourier series Additive synthesis Harmonic series (music) Computer music

2.4 Notes

2.5 References Amerio, Luigi; Prouse, Giovanni (1971), Almost-periodic functions and functional equations, The UniversitySeries in Higher Mathematics, New YorkCincinnatiTorontoLondonMelbourne: Van Nostrand Reinhold,pp. viii+184, ISBN 0-442-20295-4, MR 275061, Zbl 0215.15701.

A.S. Besicovitch, On generalized almost periodic functions Proc. London Math. Soc. (2), 25 (1926) pp.495512

A.S. Besicovitch, Almost periodic functions, Cambridge Univ. Press (1932) Bochner, S. (1926), Beitrage zur Theorie der fastperiodischen Funktionen, Math. Annalen 96: 119147,doi:10.1007/BF01209156

S. Bochner and J. von Neumann, Almost Periodic Function in a Group II, Trans. Amer. Math. Soc., 37 no.1 (1935) pp. 2150

H. Bohr, Zur Theorie der fastperiodischen Funktionen I Acta Math., 45 (1925) pp. 29127 H. Bohr, Almost-periodic functions, Chelsea, reprint (1947) Bredikhina, E.A. (2001), A/a011970, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer, ISBN978-1-55608-010-4

Bredikhina, E.A. (2001), Besicovitch almost periodic functions, in Hazewinkel, Michiel, Encyclopedia ofMathematics, Springer, ISBN 978-1-55608-010-4

Bredikhina, E.A. (2001), Bohr almost periodic functions, in Hazewinkel, Michiel, Encyclopedia of Mathe-matics, Springer, ISBN 978-1-55608-010-4

Bredikhina, E.A. (2001), Stepanov almost periodic functions, in Hazewinkel, Michiel, Encyclopedia ofMath-ematics, Springer, ISBN 978-1-55608-010-4


Bredikhina, E.A. (2001), Weyl almost periodic functions, in Hazewinkel, Michiel, Encyclopedia of Mathe-matics, Springer, ISBN 978-1-55608-010-4

J. von Neumann, Almost Periodic Functions in a Group I, Trans. Amer. Math. Soc., 36 no. 3 (1934) pp.445492

W. Stepano(=V.V. Stepanov), Sur quelques gnralisations des fonctions presque priodiques C.R. Acad.Sci. Paris, 181 (1925) pp. 9092

W. Stepano(=V.V. Stepanov), Ueber einige Verallgemeinerungen der fastperiodischen Funktionen Math.Ann., 45 (1925) pp. 473498

H. Weyl, Integralgleichungen und fastperiodische Funktionen Math. Ann., 97 (1927) pp. 338356

2.6 External links Almost periodic function (equivalent denition) at PlanetMath.org.

Chapter 3

Amenable group

In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operationon bounded functions that is invariant under translation by group elements. The original denition, in terms of anitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 underthe German name messbar (measurable in English) in response to the BanachTarski paradox. In 1949 MahlonM. Day introduced the English translation amenable, apparently as a pun.[1]

The amenability property has a large number of equivalent formulations. In the eld of analysis, the denition is interms of linear functionals. An intuitive way to understand this version is that the support of the regular representationis the whole space of irreducible representations.In discrete group theory, where G has the discrete topology, a simpler denition is used. In this setting, a group isamenable if one can say what proportion of G any given subset takes up.If a group has a Flner sequence then it is automatically amenable.

3.1 Denition for locally compact groupsLet G be a locally compact Hausdor group. Then it is well known that it possesses a unique, up-to-scale left- (orright-) rotation invariant ring measure, the Haar measure. (This is Borel regular measure whenG is second-countable;there are both left and right measures when G is compact.) Consider the Banach space L(G) of essentially boundedmeasurable functions within this measure space (which is clearly independent of the scale of the Haar measure).Denition 1. A linear functional in Hom(L(G), R) is said to be amean if has norm 1 and is non-negative, i.e.f 0 a.e. implies (f) 0.Denition 2. A mean in Hom(L(G), R) is said to be left-invariant (resp. right-invariant) if (gf) = (f) forall g in G, and f in L(G) with respect to the left (resp. right) shift action of gf(x) = f(g1x)(resp. fg(x) = f(xg1) ).Denition 3. A locally compact Hausdor group is called amenable if it admits a left- (or right-)invariant mean.

3.2 Equivalent conditions for amenabilityPier (1984) contains a comprehensive account of the conditions on a second countable locally compact group G thatare equivalent to amenability:[2]

Existence of a left (or right) invariant mean on L(G). The original denition, which depends on the axiomof choice.

Existence of left-invariant states. There is a left-invariant state on any separable left-invariant unital C*subalgebra of the bounded continuous functions on G.

Fixed-point property. Any action of the group by continuous ane transformations on a compact convexsubset of a (separable) locally convex topological vector space has a xed point. For locally compact abeliangroups, this property is satised as a result of the MarkovKakutani xed-point theorem.

9

10 CHAPTER 3. AMENABLE GROUP

Irreducible dual. All irreducible representations are weakly contained in the left regular representation onL2(G).

Trivial representation. The trivial representation of G is weakly contained in the left regular representation. Godement condition. Every bounded positive-denite measure on G satises (1) 0. Valette (1998) im-proved this criterion by showing that it is sucient to ask that, for every continuous positive-denite compactlysupported function f on G, the function f has non-negative integral with respect to Haar measure, where denotes the modular function.

Days asymptotic invariance condition. There is a sequence of integrable non-negative functions n withintegral 1 on G such that (g)n n tends to 0 in the weak topology on L1(G).

Reiters condition. For every nite (or compact) subset F of G there is an integrable non-negative function with integral 1 such that (g) is arbitrarily small in L1(G) for g in F.

Dixmiers condition. For every nite (or compact) subset F of G there is unit vector f in L2(G) such that(g)f f is arbitrarily small in L2(G) for g in F.

GlicksbergReiter condition. For any f in L1(G), the distance between 0 and the closed convex hull in L1(G)of the left translates (g)f equals |f |.

Flner condition. For every nite (or compact) subset F of G there is a measurable subset U of G with nitepositive Haar measure such that m(U gU)/m(U) is arbitrarily small for g in F.

Leptins condition. For every nite (or compact) subset F ofG there is a measurable subsetU ofG with nitepositive Haar measure such that m(FU U)/m(U) is arbitrarily small.

Kestens condition. Left convolution on L2(G) by a symmetric probability measure on G gives an operator ofoperator norm 1.

Johnsons cohomological condition. The Banach algebra A = L1(G) is amenable as a Banach algebra, i.e.any bounded derivation of A into the dual of a Banach A-bimodule is inner.

3.3 Case of discrete groupsThe denition of amenability is simpler in the case of a discrete group,[3] i.e. a group equipped with the discretetopology.[4]

Denition. A discrete group G is amenable if there is a nitely additive measure (also called a mean) a functionthat assigns to each subset of G a number from 0 to 1such that

1. The measure is a probability measure: the measure of the whole group G is 1.2. The measure is nitely additive: given nitely many disjoint subsets of G, the measure of the union of the sets

is the sum of the measures.3. The measure is left-invariant: given a subset A and an element g of G, the measure of A equals the measure

of gA. (gA denotes the set of elements ga for each element a in A. That is, each element of A is translated onthe left by g.)

This denition can be summarized thus: G is amenable if it has a nitely-additive left-invariant probability measure.Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a randomelement of G is in A?It is a fact that this denition is equivalent to the denition in terms of L(G).Having a measure on G allows us to dene integration of bounded functions on G. Given a bounded function f : G R, the integral

ZG

f d

3.4. PROPERTIES 11

is dened as in Lebesgue integration. (Note that some of the properties of the Lebesgue integral fail here, since ourmeasure is only nitely additive.)If a group has a left-invariant measure, it automatically has a bi-invariant one. Given a left-invariant measure , thefunction (A) = (A1) is a right-invariant measure. Combining these two gives a bi-invariant measure:

(A) =

Zg2G

Ag1

d:

The equivalent conditions for amenability also become simpler in the case of a countable discrete group . For sucha group the following conditions are equivalent:[5]

is amenable.

If acts by isometries on a (separable) Banach space E, leaving a weakly closed convex subset C of the closedunit ball of E* invariant, then has a xed point in C.

There is a left invariant norm-continuous functional on () with (1) = 1 (this requires the axiom ofchoice).

There is a left invariant state on any left invariant separable unital C* subalgebra of ().

There is a set of probability measures n on such that ||g n n||1 tends to 0 for each g in (M.M. Day).

There are unit vectors xn in 2() such that ||g xn xn||2 tends to 0 for each g in (J. Dixmier).

There are nite subsets Sn of such that |g Sn Sn| / |Sn| tends to 0 for each g in (Flner).

If is a symmetric probability measure on with support generating , then convolution by denes anoperator of norm 1 on 2() (Kesten).

If acts by isometries on a (separable) Banach space E and f in (, E*) is a bounded 1-cocycle, i.e. f(gh)= f(g) + gf(h), then f is a 1-coboundary, i.e. f(g) = g for some in E* (B.E. Johnson).

The von Neumann group algebra of is hypernite (A. Connes).

Note that A. Connes also proved that the von Neumann group algebra of any connected locally compact group ishypernite, so the last condition no longer applies in the case of connected groups.Amenability is related to the spectral problem of Laplacians. For instance, the fundamental group of a closed Rie-mannian manifold is amenable if and only if the bottom of the spectrum of the Laplacian is 0 (R. Brooks, T. Sunada).

3.4 Properties Every (closed) subgroup of an amenable group is amenable.

Every quotient of an amenable group is amenable.

A group extension of an amenable group by an amenable group is again amenable. In particular, nite directproduct of amenable groups are amenable, although innite products need not be.

Direct limits of amenable groups are amenable. In particular, if a group can be written as a directed union ofamenable subgroups, then it is amenable.

Amenable groups are unitarizable; the converse is an open problem.

Countable discrete amenable groups obey the Ornstein isomorphism theorem.[6][7]


3.5 Examples

Finite groups are amenable. Use the counting measure with the discrete denition. More generally, compactgroups are amenable. The Haar measure is an invariant mean (unique taking total measure 1).

The group of integers is amenable (a sequence of intervals of length tending to innity is a Flner sequence).The existence of a shift-invariant, nitely additive probability measure on the group Z also follows easily fromthe HahnBanach theorem this way. Let S be the shift operator on the sequence space (Z), which is denedby (Sx)i = xi for all x (Z), and let u (Z) be the constant sequence ui = 1 for all i Z. Any element y Y:=Ran(S I) has a distance larger than or equal to 1 from u (otherwise yi = xi+1 - xi would be positive andbounded away from zero, whence xi could not be bounded). This implies that there is a well-dened norm-onelinear form on the subspace Ru + Y taking tu + y to t. By the HahnBanach theorem the latter admits a norm-one linear extension on (Z), which is by construction a shift-invariant nitely additive probability measureon Z.

If every conjugacy class in a locally compact group has compact closure, then the group is amenable. Examplesof groups with this property include compact groups, locally compact abelian groups, and discrete groups withnite conjugacy classes.[8]

By the direct limit property above, a group is amenable if all its nitely generated subgroups are. That is, locallyamenable groups are amenable.

By the fundamental theorem of nitely generated abelian groups, it follows that abelian groups areamenable.

It follows from the extension property above that a group is amenable if it has a nite index amenable subgroup.That is, virtually amenable groups are amenable.

Furthermore, it follows that all solvable groups are amenable.

All examples above are elementary amenable. The rst class of examples below can be used to exhibit non-elementaryamenable examples thanks to the existence of groups of intermediate growth.

Finitely generated groups of subexponential growth are amenable. A suitable subsequence of balls will providea Flner sequence.[9]

Finitely generated innite simple groups cannot be obtained by bootstrap constructions as used to constructelementary amenable groups. Since there exist such simple groups that are amenable, due to Juschenko andMonod,[10] this provides again non-elementary amenable examples.

3.6 CounterexamplesIf a countable discrete group contains a (non-abelian) free subgroup on two generators, then it is not amenable. Theconverse to this statement is the so-called von Neumann conjecture, which was disproved by Olshanskii in 1980 usinghis Tarski monsters. Adyan subsequently showed that free Burnside groups are non-amenable: since they are periodic,they cannot contain the free group on two generators. These groups are nitely generated, but not nitely presented.However, in 2002 Sapir and Olshanskii found nitely presented counterexamples: non-amenable nitely presentedgroups that have a periodic normal subgroup with quotient the integers.[11]

For nitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative:[12] everysubgroup of GL(n,k) with k a eld either has a normal solvable subgroup of nite index (and therefore is amenable)or contains the free group on two generators. Although Tits' proof used algebraic geometry, Guivarc'h later foundan analytic proof based on V. Oseledets' multiplicative ergodic theorem.[13] Analogues of the Tits alternative havebeen proved for many other classes of groups, such as fundamental groups of 2-dimensional simplicial complexes ofnon-positive curvature.[14]

3.7. SEE ALSO 13

3.7 See also

Uniformly bounded representation

Kazhdans property (T)

Von Neumann conjecture

3.8 Notes

[1] Days rst published use of the word is in his abstract for an AMS summer meeting in 1949, Means on semigroups andgroups, Bull. A.M.S. 55 (1949) 10541055. Many text books on amenabilty, such as Volker Rundes, suggest that Daychose the word as a pun.

[2] Pier 1984

[3] See:

Greenleaf 1969 Pier 1984 Takesaki 2002a Takesaki 2002b

[4] Weisstein, Eric W., Discrete Group, MathWorld.

[5] Pier 1984

[6] Ornstein, D.; Weiss, B. (1987). Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math.48: 1141.

[7] Lewis Bowen (2011), "Every countably innite group is almost Ornstein", ArXiv abs/1103.4424

[8] Leptin 1968

[9] See:

Greenleaf 1969 Pier 1984 Takesaki 2002a Takesaki 2002b

[10] Juschenko, Kate; Monod, Nicolas (2013), Cantor systems, piecewise translations and simple amenable groups, Annalsof Mathematics 178 (2): 775787, doi:10.4007/annals.2013.178.2.7

[11] Olshanskii, Alexander Yu.; Sapir, MarkV. (2002), Non-amenable nitely presented torsion-by-cyclic groups, Publ. Math.Inst. Hautes tudes Sci. 96: 43169, doi:10.1007/s10240-002-0006-7

[12] Tits, J. (1972), Free subgroups in linear groups, J. Algebra 20 (2): 250270, doi:10.1016/0021-8693(72)90058-0

[13] Guivarc'h, Yves (1990), Produits de matrices alatoires et applications aux proprits gometriques des sous-groupes dugroupes linaire, Ergod. Th. & Dynam. Sys. 10 (3): 483512, doi:10.1017/S0143385700005708

[14] Ballmann, Werner; Brin, Michael (1995), Orbihedra of nonpositive curvature, Inst. Hautes tudes Sci. Publ. Math. 82:169209, doi:10.1007/BF02698640


3.9 ReferencesThis article incorporates material from Amenable group on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

Brooks, Robert (1981), The fundamental group and the spectrum of the laplacian, Comment. Math. Helv.56: 581598, doi:10.1007/bf02566228

Dixmier, Jacques (1977), C*-algebras (translated from the French by Francis Jellett), North-Holland Mathe-matical Library 15, North-Holland

Greenleaf, F.P. (1969), InvariantMeans on Topological Groups and Their Applications, VanNostrand Reinhold Juschenko, Kate; Monod, Nicolas (2013), Cantor systems, piecewise translations and simple amenable groups,

Annals of Mathematics 178 (2): 775787, doi:10.4007/annals.2013.178.2.7 Leptin, H. (1968), Zur harmonischen Analyse klassenkompakter Gruppen, Invent. Math. 5: 249254,doi:10.1007/bf01389775

Pier, Jean-Paul (1984),Amenable locally compact groups, Pure andAppliedMathematics,Wiley, Zbl 0621.43001 Runde, V. (2002), Lectures on Amenability, LectureNotes inMathematics 1774, Springer, ISBN9783540428527 Sunada, Toshikazu (1989), Unitary representations of fundamental groups and the spectrum of twisted Lapla-cians, Topology 28: 125132, doi:10.1016/0040-9383(89)90015-3

Takesaki, M. (2002a), Theory of Operator Algebras 2, Springer, ISBN 9783540422488 Takesaki, M. (2002b), Theory of Operator Algebras 3, Springer, ISBN 9783540429142

Valette, Alain (1998), On Godements characterisation of amenability, Bull. Austral. Math. Soc. 57: 153158, doi:10.1017/s0004972700031506

von Neumann, J (1929), Zur allgemeinen Theorie des Maes (PDF), Fund. Math. 13 (1): 73111

3.10 External links Some notes on amenability by Terry Tao

Chapter 4

Basic subgroup

In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroupsand satises further technical conditions. This notion was introduced by L. Ya. Kulikov (for p-groups) and by LszlFuchs (in general) in an attempt to formulate classication theory of innite abelian groups that goes beyond thePrfer theorems. It helps to reduce the classication problem to classication of possible extensions between twowell understood classes of abelian groups: direct sums of cyclic groups and divisible groups.

4.1 Denition and propertiesA subgroup B of an abelian group A is called p-basic, for a xed prime number p, if the following conditions hold:

(1) B is a direct sum of cyclic groups of order pn and innite cyclic groups;(2) B is a p-pure subgroup of A;(3) The quotient group A/B is a p-divisible group.

Conditions (1) (3) imply that the subgroup B is Hausdor in the p-adic topology of B, which moreover coincideswith the topology induced from A, and that B is dense in A. Picking a generator in each cyclic direct summand of Bcreates a p-basis of B, which is analogous to a basis of a vector space or a free abelian group.Every abelian group A contains p-basic subgroups for each p, and any two p-basic subgroups of A are isomorphic.Abelian groups that contain a unique p-basic subgroup have been completely characterized. For the case of p-groupsthey are either divisible or bounded, i.e. have bounded exponent. In general, the isomorphism class of the quotientA/B by a basic subgroup B may depend on B.

4.2 Generalization to modulesThe notion of a p-basic subgroup in an abelian p-group admits a direct generalization to modules over a principalideal domain. The existence of such a basic submodule and uniqueness of its isomorphism type continue to hold.

4.3 References Lszl Fuchs (1970), Innite abelian groups, Vol. I. Pure and Applied Mathematics, Vol. 36. New YorkLondon: Academic Press MR 0255673

L. Ya. Kulikov, On the theory of abelian groups of arbitrary cardinality (in Russian), Math. Sb., 16 (1945),129162

Kurosh, A. G. (1960), The theory of groups, New York: Chelsea, MR 0109842

15

Chapter 5

Bohr compactication

In mathematics, theBohr compactication of a topological groupG is a compact Hausdor topological groupH thatmay be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodicfunctions on G to the theory of continuous functions on H. The concept is named after Harald Bohr who pioneeredthe study of almost periodic functions, on the real line.

5.1 Denitions and basic propertiesGiven a topological group G, the Bohr compactication of G is a compact Hausdor topological group Bohr(G)and a continuous homomorphism

b: G Bohr(G)

which is universal with respect to homomorphisms into compact Hausdor groups; this means that if K is anothercompact Hausdor topological group and

f: G K

is a continuous homomorphism, then there is a unique continuous homomorphism

Bohr(f): Bohr(G) K

such that f = Bohr(f) b.Theorem. The Bohr compactication exists and is unique up to isomorphism.We will denote the Bohr compactication of G by Bohr(G) and the canonical map by

b : G! Bohr(G):

The correspondence G Bohr(G) denes a covariant functor on the category of topological groups and continuoushomomorphisms.The Bohr compactication is intimately connected to the nite-dimensional unitary representation theory of a topo-logical group. The kernel of b consists exactly of those elements of G which cannot be separated from the identityof G by nite-dimensional unitary representations.The Bohr compactication also reduces many problems in the theory of almost periodic functions on topologicalgroups to that of functions on compact groups.A bounded continuous complex-valued function f on a topological group G is uniformly almost periodic if andonly if the set of right translates gf where

16

5.2. MAXIMALLY ALMOST PERIODIC GROUPS 17

[gf ](x) = f(g1 x)

is relatively compact in the uniform topology as g varies through G.Theorem. A bounded continuous complex-valued function f on G is uniformly almost periodic if and only if thereis a continuous function f1 on Bohr(G) (which is uniquely determined) such that

f = f1 b:

5.2 Maximally almost periodic groupsTopological groups for which the Bohr compactication mapping is injective are calledmaximally almost periodic (orMAP groups). In the case G is a locally compact connected group, MAP groups are completely characterized: Theyare precisely products of compact groups with vector groups of nite dimension.

5.3 References Hazewinkel, Michiel, ed. (2001), B/b016780, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 6

Cantor cube

In mathematics, a Cantor cube is a topological group of the form {0, 1}A for some index set A. Its algebraic andtopological structures are the group direct product and product topology over the cyclic group of order 2 (which isitself given the discrete topology).If A is a countably innite set, the corresponding Cantor cube is a Cantor space. Cantor cubes are special amongcompact groups because every compact group is a continuous image of one, although usually not a homomorphicimage. (The literature can be unclear, so for safety, assume all spaces are Hausdor.)Topologically, any Cantor cube is:

homogeneous; compact; zero-dimensional; AE(0), an absolute extensor for compact zero-dimensional spaces. (Every map from a closed subset of such aspace into a Cantor cube extends to the whole space.)

By a theorem of Schepin, these four properties characterize Cantor cubes; any space satisfying the properties ishomeomorphic to a Cantor cube.In fact, every AE(0) space is the continuous image of a Cantor cube, and with some eort one can prove that everycompact group is AE(0). It follows that every zero-dimensional compact group is homeomorphic to a Cantor cube,and every compact group is a continuous image of a Cantor cube.

6.1 References Todorcevic, Stevo (1997). Topics in Topology. ISBN 3-540-62611-5. A.A. Mal'tsev (2001), Colon, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

18

Chapter 7

Chabauty topology

In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty,on the set of all closed subgroups of a locally compact group G.The intuitive idea may be seen in the case of the set of all lattices in a Euclidean space E. There these are only certainof the closed subgroups: others can be found by in a sense taking limiting cases or degenerating a certain sequence oflattices. One can nd linear subspaces or discrete groups that are lattices in a subspace, depending on how one takesa limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology.This topology can be derived from the Vietoris topology construction, a topological structure on all non-empty subsetsof a space. More precisely, it is an adaptation of the Fell topology construction, which itself derives from the Vietoristopology concept.

7.1 References Claude Chabauty, Limite d'ensembles et gomtrie des nombres. Bulletin de la SocitMathmatique de France,78 (1950), p. 143-151

19

Chapter 8

Circle group

For the jazz group, see Circle (jazz band).

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolutevalue 1, i.e., the unit circle in the complex plane or simply the unit complex numbers[1]

T = fz 2 C : jzj = 1g:

The circle group forms a subgroup of C, the multiplicative group of all nonzero complex numbers. Since C isabelian, it follows that T is as well. The circle group is also the group U(1) of 11 unitary matrices; these act on thecomplex plane by rotation about the origin. The circle group can be parametrized by the angle of rotation by

7! z = ei = cos + i sin :

This is the exponential map for the circle group.The circle group plays a central role in Pontryagin duality, and in the theory of Lie groups.The notation T for the circle group stems from the fact t

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