1. Conjugacy and Dynamics in Thompson’s Groups Jim Belk (joint with Francesco Matucci)

2. Thompson’s Group Thompson’s Group  . 2 ½ 1 (¼,½)(¼,½) (½,¾)   Piecewise-linear homeomorphisms of  .   if and only if: 1. The slopes of  are powers of 2, and 2. The breakpoints of  have dyadic rational coordinates.

3. Thompson’s Group  . 2 ½ 1 (¼,½)(¼,½) (½,¾)   if and only if: 1. The slopes of  are powers of 2, and 2. The breakpoints of  have dyadic rational coordinates. 10 ½              ¼ 01 ½ ¾

4. Another Example                          

5. Another Example                          

6. Another Example                          

7. Another Example                          

8. Another Example                          

9. Another Example                          

10. Another Example                          

11. Another Example In general, a dyadic subdivision is any subdivision of  obtained by repeatedly cutting intervals in half. Every element of maps linearly between the intervals of two dyadic subdivisions.

12. Properties of is infinite and torsion-free. is finitely presented. has an Eilenberg-Maclane complex with two cells in each dimension (finiteness property  ).   is simple, and      . has exponential growth, but does not contain a free group. may or may not be amenable.

13. Other Groups is one of three Thompson groups. The other two are:

14. Other Groups is one of three Thompson groups. The other two are: Thompson’s group  (similar to but on a circle).                                

15. Other Groups is one of three Thompson groups. The other two are: Thompson’s group  (similar to but on a circle). Thompson’s group  (similar to but not continuous). 10 ½              ¼ 01 ½ ¾                                

16. The Conjugacy Problem A solution to the conjugacy problem in  is an algorithm which decides whether given elements     are conjugate:     Let  be any group.

17. Higman (1974): Shows that  has solvable conjugacy problem. Mather (1974): Describes conjugacy in Diff  . Brin and Squier (1987): Describe conjugacy in  . Guba and Sapir (1997): Solve the conjugacy problem in. B and Matucci (2006): Unified solution to the conjugacy problems in, , and . A solution to the conjugacy problem in  is an algorithm which decides whether given elements     are conjugate:    

18. Strand Diagrams

19. We represent elements of using strand diagrams: 1 0 ½              ¼ 01 ½ ¾

20. Strand Diagrams A strand diagram is similar to an automaton. It takes a number     (expressed in binary) as input, and outputs .  

21. Strand Diagrams A strand diagram is similar to an automaton. It takes a number     (expressed in binary) as input, and outputs . 

22. Strand Diagrams A strand diagram is similar to an automaton. It takes a number     (expressed in binary) as input, and outputs .  

23. Strand Diagrams A strand diagram is similar to an automaton. It takes a number     (expressed in binary) as input, and outputs .   

24. Strand Diagrams A strand diagram is similar to an automaton. It takes a number     (expressed in binary) as input, and outputs .    

25.  Strand Diagrams A strand diagram is similar to an automaton. It takes a number     (expressed in binary) as input, and outputs .    

26. Strand Diagrams Every vertex (other than the top and the bottom) is either a split or a merge: split merge

27. Strand Diagrams A split removes the first digit of a binary expansion: merge      

28.  Strand Diagrams A merge inserts a new digit:          

29. Strand Diagrams 1 0 ½              ¼ 01 ½ ¾

30. Strand Diagrams 1 0 ½              ¼ 01 ½ ¾     

31. Strand Diagrams 1 0 ½              ¼ 01 ½ ¾         

32. Strand Diagrams 1 0 ½              ¼ 01 ½ ¾       

33. Strand Diagrams

41. General Definition: 1.Acyclic directed graph 2.Embedded in the square 3.One source and one sink 4.Other vertices are splits and merges Theorem. Every strand diagram represents an element of Thompson’s group.

42. Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.

43. Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.   

44. Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.   

45. Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.    

46. Reduction Type I Type II These two moves are called reductions. Neither affects the corresponding piecewise-linear function.    

47. Reduction Type I Type II A strand diagram is reduced if it is not subject to any reductions. Theorem. There is a one-to-one correspondence: reduced strand diagrams elements of

48. Multiplication We can multiply two strand diagrams concatenating them:       

49. Multiplication Usually the result will not be reduced.       

50. Multiplication Usually the result will not be reduced.       

51. Multiplication Usually the result will not be reduced.       

52. Multiplication Usually the result will not be reduced.       

53. Multiplication Usually the result will not be reduced.       

54. Multiplication Usually the result will not be reduced.       

55. Multiplication Usually the result will not be reduced.        reduced

56. Conjugacy

57. Motivation Here’s a solution to the conjugacy problem in the free group  . Suppose we are given a reduced word:           To find the conjugacy class, make the word into a circle and reduce:

58. Motivation Here’s a solution to the conjugacy problem in the free group  . Suppose we are given a reduced word:           To find the conjugacy class, make the word into a circle and reduce:             

59. Motivation To find the conjugacy class, make the word into a circle and reduce:              Two elements of   are conjugate if and only if they have the same reduced circle. Note: Two reduced circles can be compared in linear time (even though the obvious algorithm is quadratic.)

60. The Solution for The idea is to wrap the strand diagram around in a circle: We call this an annular strand diagram.

61. The Solution for The idea is to wrap the strand diagram around in a circle: We call this an annular strand diagram.

62. The Solution for The idea is to wrap the strand diagram around in a circle: We call this an annular strand diagram.

63. The Solution for The idea is to wrap the strand diagram around in a circle: We call this an annular strand diagram.

64. The Solution for The idea is to wrap the strand diagram around in a circle: We call this an annular strand diagram.

65. The Solution for The idea is to wrap the strand diagram around in a circle: We call this an annular strand diagram.

66. The Solution for The idea is to wrap the strand diagram around in a circle: We call this an annular strand diagram.

67. The Solution for The idea is to wrap the strand diagram around in a circle: We call this an annular strand diagram.

68. The Solution for Definition. An annular strand diagram is a directed graph on the annulus such that: 1. Every vertex is a split or a merge: 2. Every directed loop has positive index around the central hole. Fact. Every annular strand diagram can be obtained by “wrapping” a strand diagram.

69. Reductions There are three types of reductions. Theorem. Every annular strand diagram is equivalent to a unique reduced annular strand diagram. Type I Type II Type III

70. Theorem (Guba and Sapir). Two elements of are conjugate if and only if they have the same reduced annular strand diagram. The Solution for

71. Theorem (Guba and Sapir). Two elements of are conjugate if and only if they have the same reduced annular strand diagram. Hopcroft and Wong (1974): You can determine whether two planar graphs are isomorphic in linear time. Theorem (B and Matucci). The conjugacy problem in has a linear-time solution. The Solution for

72. Conjugacy in  and 

73. Thompson’s Group   is similar to, but it acts on the circle.                   

74. Thompson’s Group               Elements of  can be represented by strand diagrams on a cylinder.      

75. Thompson’s Group               Elements of  can be represented by strand diagrams on a cylinder.      

76. Thompson’s Group               Elements of  can be represented by strand diagrams on a cylinder.      

77. Thompson’s Group               Elements of  can be represented by strand diagrams on a cylinder.      

78. Thompson’s Group  Wrapping the cylinder around gives a strand diagram on a torus. Note: Special cutting class in  .

79. Thompson’s Group  Theorem (B and Matucci). Two elements of  are conjugate if and only if they have the same reduced toral strand diagram. Note: Special cutting class in  .

80. Thompson’s Group  Thompson’s group  is similar to, but the functions are not required to be continuous. 10 ½              ¼ 0 1 ½ ¾

81. Thompson’s Group  Thompson’s group  is similar to, but the functions are not required to be continuous. An abstract strand diagram is any directed, acyclic graph whose vertices are merges and splits (plus an input and an output).

82. Thompson’s Group  When you close an abstract strand diagram, you must keep track of the cutting cohomology class. Theorem (B and Matucci). Two elements of  are conjugate if and only if they have the same reduced closed abstract strand diagram.

83. Brin and Squier

84. Conjugacy in    Brin and Squier (1986) gave a dynamical description of conjugacy in the group   .

85. Conjugacy in    Brin and Squier (1986) gave a dynamical description of conjugacy in the group   . The first step is to break a function into its components: Each   is a one-bump function.               

86. Conjugacy in    Brin and Squier (1986) gave a dynamical description of conjugacy in the group   . Theorem.       are conjugate if and only if: 1.They have the same number of components, and 2.   is conjugate to   for each .   

87. Conjugacy in    Brin and Squier (1986) gave a dynamical description of conjugacy in the group   . Next, Brin and Squier determined conjugacy for one-bump functions. Specifically, they found a piecewise- linear version of the Mather invariant.

89. Dynamics of

90. The relation between conjugacy and dynamics in is more subtle.   Elements of can have both dyadic and non-dyadic fixed points.

91. Dynamics of The relation between conjugacy and dynamics in is more subtle.   The components of an element   are the portions between the dyadic fixed points.  

92. Dynamics of The relation between conjugacy and dynamics in is more subtle.    

93. Dynamics of The relation between conjugacy and dynamics in is more subtle.   The components of an element   are the portions between the dyadic fixed points. Theorem. Two elements of are conjugate if and only if they have conjugate components.  

94. Connected Diagrams The main structural feature of a connected annular strand diagram are its directed loops.

95. Connected Diagrams The main structural feature of a connected annular strand diagram are its directed loops. If the diagram is reduced, a loop cannot have both splits and merges.

96. The main structural feature of a connected annular strand diagram are its directed loops. If the diagram is reduced, a loop cannot have both splits and merges. So there are split loops, Connected Diagrams

97. The main structural feature of a connected annular strand diagram are its directed loops. If the diagram is reduced, a loop cannot have both splits and merges. So there are split loops, and merge loops. Connected Diagrams

98. Directed Loops Each directed loop indicates a fixed point. 

99. Directed Loops Each directed loop indicates a fixed point.   

100. Directed Loops Each directed loop indicates a fixed point.     

101. Directed Loops Each directed loop indicates a fixed point.       

102. Directed Loops Each directed loop indicates a fixed point.         

103.          Elements of this conjugacy class have an attracting fixed point, with binary expansion:      

104. Directed Loops A split loop gives a repelling fixed point:    

105. Directed Loops Each directed loop indicates a fixed point.        

106. Directed Loops Each directed loop indicates a fixed point.            

107. Directed Loops Each directed loop indicates a fixed point.            

108. Directed Loops A split loop gives a repelling fixed point:           

109. Directed Loops Each directed loop indicates a fixed point.          

110. Directed Loops Each directed loop indicates a fixed point.         

111. Directed Loops Each directed loop indicates a fixed point.        

112. Directed Loops A split loop gives a repelling fixed point:       

113. Directed Loops A split loop gives a repelling fixed point:        

114. Any element in this conjugacy class has exactly four fixed points:

115. The material between the loops determines the three “bumps”.

116. For  and  Directed loops in  and  are periodic points. Theorem. Any element of  or  has a periodic point. Corollary (Ghys). Every element of  has rational rotation number. Theorem. Any two   subgroups of  are conjugate. Theorem. The conjugacy class of a torsion element in  is determined solely by the set of periods.