1. Forest Diagrams for Thompson’s Group Jim Belk

2. Associative Laws Consider the following piecewise-linear homeomorphism of  :

3. Associative Laws This homeomorphism is called the basic associative law. It corresponds to the operation        

4. Associative Laws Here’s a different associative law. It corresponds to   .

5. Associative Laws           General Definition: A dyadic subdivision of    is obtained by repeatedly cutting intervals in half:

6. Associative Laws An associative law is a PL-homeomorphism that maps linearly between the intervals of two dyadic subdivisions.

7. Associative Laws                  

8. Associative Laws Thompson’s Group is the group of all associative laws.

9. Associative Laws Thompson’s Group is the group of all associative laws. If  , then: Every slope of  is a power of 2. Every breakpoint of  has dyadic rational coordinates. The converse also holds. 2 ½ 1 (¼,½)(¼,½) (½,¾)

10. Properties of is an infinite, torsion-free group.

11. Properties of is an infinite, torsion-free group. is finitely generated:  

12. Properties of is an infinite, torsion-free group. is finitely generated. is finitely presented:       2 relations 

13. Properties of is an infinite, torsion-free group. is finitely generated. is finitely presented. admits a   complex with exactly two cells in each dimension.

14. Properties of    is simple. Every proper quotient of is abelian. has exponential growth. contains       . does not contain  . Is amenable?

15. Tree Diagrams

16. We can represent an associative law by a pair of binary trees: This is called a tree diagram.

17. Tree Diagrams Unfortunately, the tree diagram for an element of is not unique.

18. Tree Diagrams Unfortunately, the tree diagram for an element of is not unique.

19. Tree Diagrams Unfortunately, the tree diagram for an element of is not unique.

20. Tree Diagrams Unfortunately, the tree diagram for an element of is not unique. We can always cancel opposing pairs of carets. This is called a reduction of the tree diagram.

21. Multiplying Tree Diagrams 

22. 

23. Multiplying Tree Diagrams 

24. 

25. Generators Here are the tree diagrams for the generators:  

26. The Action on 

27. If we conjugate by the homeomorphism:   we get an action of on .

28. A dyadic subdivision of  is obtained by repeatedly cutting intervals in half:  is the group of PL-homeomorphisms of  that map linearly between the intervals of two dyadic subdivisions The Action on 

29.     What’s the point? The generators become simpler. Here’s the new picture for   :

30. The Action on      What’s the point? The generators become simpler. And here’s   :

31. Forest Diagrams

32. We can represent an element of using a pair of binary forests: 

33. Forest Diagrams This is called a forest diagram. 

34. Forest Diagrams This is called a forest diagram.                   

35.   Here are the forest diagrams for the generators:

36. The Action of   Left-multiplication by   moves the top pointer of a forest diagram: 

37. The Action of   Left-multiplication by   moves the top pointer of a forest diagram: 

38. The Action of   Left-multiplication by   moves the top pointer of a forest diagram: 

39. The Action of   Left-multiplication by   moves the top pointer of a forest diagram: 

40. The Action of   Left-multiplication by   moves the top pointer of a forest diagram: 

41. The Action of   Left-multiplication by   moves the top pointer of a forest diagram: 

42.   Here are the forest diagrams for the generators:

43. The Action of   Left-multiplication by   adds a new caret on the top: 

44. The Action of   Left-multiplication by   adds a new caret on the top: 

45. The Action of   Left-multiplication by   adds a new caret on the top: 

46. The Action of   Left-multiplication by   adds a new caret on the top: 

47. The Action of   Sometimes the new caret appears opposite a caret on the bottom: 

48. The Action of   Sometimes the new caret appears opposite a caret on the bottom: 

49. The Action of   Sometimes the new caret appears opposite a caret on the bottom:

50. The Action of   So   can delete bottom carets (and    can create bottom carets.)

51. Lengths

52. Finding Lengths Problem. Given an  , find the       - length of . Example. Find the length of:

53. Finding Lengths

54. 1

55. 1

56. 2

57. 21

58. 211

59. 211

60. 2111

61. 2121

62. 2221

63. 2221 (1 + 2 + 2 + 2) + 2 = 9  This element has length 9.

64. Example 2 Example. Find the length of:

65. Example 2

66. 1

67. 11

68. 11

69. 111

70. 1111

71. 1111

72. 1112

73. 1122

74. 1222

75. 2222

76. 2222 (2 + 2 + 2 + 2) + 2 = 10  This element has length 10.

77. Example 3 Example. Find the length of:

78. Example 3

79. 1

80. 11

81. 11

82. 111

83. 111

84. 112

85. 112

86. 122

87. 122

88. 222

89. 2221

90. 2221 (1 + 2 + 2 + 2) + 4 = 11  This element has length 11.

91. Labeling. Label each space as follows. The Length Formula

92. Labeling. Label each space as follows. L:Left of the pointer, exterior. The Length Formula L

93. Labeling. Label each space as follows. L:Left of the pointer, exterior. N:Next to a caret on the left. The Length Formula LNNN

94. Labeling. Label each space as follows. L:Left of the pointer, exterior. N:Next to a caret on the left. R:Right of the pointer, exterior. The Length Formula LNNNR

95. Labeling. Label each space as follows. L:Left of the pointer, exterior. N:Next to a caret on the left. R:Right of the pointer, exterior. I:Interior The Length Formula LNRNNIII

96. Theorem. The weight of a space is determined by its label pair. LNRI L2111 N1222 R1220 I1200 L:Left exterior. N:Next to a caret. R:Right exterior. I:Interior

97. LNRI L2111 N1222 R1220 I1200 L:Left exterior. N:Next to a caret. R:Right exterior. I:Interior

98. LNRI L2111 N1222 R1220 I1200 L:Left exterior. N:Next to a caret. R:Right exterior. I:Interior L

99. LNRI L2111 N1222 R1220 I1200 L:Left exterior. N:Next to a caret. R:Right exterior. I:Interior N LN N

100. LNRI L2111 N1222 R1220 I1200 L:Left exterior. N:Next to a caret. R:Right exterior. I:Interior L R N NNRR RRRR

101. LNRI L2111 N1222 R1220 I1200 L:Left exterior. N:Next to a caret. R:Right exterior. I:Interior LN NNII I RRR RRRR

102. LNRI L2111 N1222 R1220 I1200 L:Left exterior. N:Next to a caret. R:Right exterior. I:Interior LN NN 1 RRR RRRRI II 202020

103. LN NNRRR RRRRI II 1202020  length     weights  +  # of carets   7 + 4  11

104. Convexity

105. A group  is convex if    is convex for each . id   

106. Convexity  is almost convex if any two elements in    a distance two apart are connected by a path of length  . id      

107. Convexity Theorem (Cleary and Taback). is not almost convex (using       ).

108. Convexity Theorem (Cleary and Taback). is not almost convex (using       ). 112222220 Length  16

109. Convexity Theorem (Cleary and Taback). is not almost convex (using       ). 122222220 Length  17

110. Convexity Theorem (Cleary and Taback). is not almost convex (using       ). 022222220 Length  16

111. Convexity Theorem (Cleary and Taback). is not almost convex (using       ). 122222220 Length  17

112. Convexity Theorem (Cleary and Taback). is not almost convex (using       ). 112222220 Length  16

113. Convexity Theorem (Cleary and Taback). is not almost convex (using       ).

114. Convexity Theorem (Cleary and Taback). is not almost convex (using       ). This gives us two elements of     a distance two apart that have distance   in    .

115. Convexity Theorem (Belk and Bux). For  even, there exist elements        such that:    , and The shortest path in     from  to  has length . Kai-Uwe and I have proven the following:

116. Amenability

117. The Isoperimetric Constant Let  be the Cayley graph of a group . If  is a finite subset of , its boundary consists of all edges between  and  .

118. The Isoperimetric Constant Let  be the Cayley graph of a group . The isoperimetric constant is:     is amenable if    .

119. Theorem (Belk and Brown).     . Proof: Let   be all elements of the form:       We claim that:

120.   Given a random    , we must compute:                                        

121.      exits    the current tree of  is trivial. Claim. Given a random element of   :   current tree is trivial    as    .      

122.   Theorem. As   , the probability  that the current tree is trivial satisfies:                     where   is the number of binary trees with  leaves.      

123. Theorem. As   , the probability  that the current tree is trivial satisfies:                     where   is the number of binary trees with  leaves. The coefficients   are the Catalan numbers, and have growth rate . The polynomial above “converges” to the generating function for the Catalan numbers, which has a vertical asymptote at   .        

124. The End