Conjugacy in Thompson’s Group 

1. Conjugacy inThompson’s Group  Jim Belk (joint with Francesco Matucci)

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

3. ½ Thompson’s Group   . 1    if and only if: 1. The slopes of  are powers of 2, and 2. The breakpoints of  have dyadic rational coordinates. (½,¾)          (¼,½) ½ 0 ¼ 1 2 0 ¾ 1 ½

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. Strand Diagrams

13.          Strand Diagrams We represent elements of  using strand diagrams: ½ 1 0 ¼ 0 ¾ 1 ½

14. Strand Diagrams  A strand diagram takes a number   (expressed in binary) as input, and outputs . 

15. Strand Diagrams  A strand diagram takes a number   (expressed in binary) as input, and outputs .

16.  Strand Diagrams  A strand diagram takes a number   (expressed in binary) as input, and outputs .

17.   Strand Diagrams  A strand diagram takes a number   (expressed in binary) as input, and outputs .

18.    Strand Diagrams  A strand diagram takes a number   (expressed in binary) as input, and outputs .

19.     Strand Diagrams  A strand diagram takes a number   (expressed in binary) as input, and outputs .

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

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

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

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

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

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

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

27. Strand Diagrams

28. Strand Diagrams

29. Strand Diagrams

30. Strand Diagrams

31. Strand Diagrams

32. Strand Diagrams

33. Strand Diagrams

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

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

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

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

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

39. Reduction A strand diagram is reducedif it is not subject to any reductions. Theorem. There is a one-to-one correspondence: Type I reducedstrand diagrams elements of  Type II

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

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

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

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

44. Conjugacy

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

46. The Free Group 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:              

47. To find the conjugacy class, make the word into a circle and reduce:               The Free Group Two elements of   are conjugate if and only if they have the same reduced circle.

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

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

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