Twisted conjugacy in braid groups

1. Twisted conjugacy in braid groups Juan González-Meneses Universidad de Sevilla Joint with E. Ventura. Tresses à Paris Rencontres Parisiennes du GDR Tresses Paris, 17-20 september 2008.

2. Introduction Conjugacy problem In a group G: a ~ b Conjugation: Conjugacy Decision Problem: Determine whether two elements are conjugate. Conjugacy Search Problem: Find a conjugating element for two given conjugate elements.

3. Introduction Twisted conjugacy problem In a group G: Reidemeister (1936) Conjugation: Twisted Fixed automorphism. Twisted Conjugacy Decision Problem: Determine whether two elements are twisted conjugate. Twisted Conjugacy Search Problem: Find a conjugating element for two given twisted conjugate elements.

4. Introduction Twisted conjugacy problem If f : G G is an innerautomorphism: Twisted conjugacy problem Conjugacy problem Just need to focus on representatives of Out(G) =Aut(G)/Inn(G).

5. Solvable conj. problem. Solvable twisted conj. problem. Introduction Motivation Bogopolski, Martino, Ventura, 2008. ? F = H = … … f.g.-abelian f.g.-free f.g.-free f.g.-t.f.-hyperbolic G has solvable conj. problem ,AG < Aut (F ) is orbit decidable One can determine, given x,y2F, whether x ~ f(y), for some f2AG. Can we put braid groups here? (Out Bn is finite)

6. Left normal form: Braid groups Normal form Bn: Braid group on n strands. Each factor is a simple element. (permutation braid) Canonical length = No. of factors.

7. Braid groups Automorphisms Automorphisms ofBn: (Dyer-Grossman, 1981) Just need to solve the twisted conjugacy problem for e.

8. Braid groups Twisted conjugacy Twisted conjugation for e: ( c written backwards ) c

9. Braid groups Twisted conjugacy Twisted conjugation for e: This is the twisted conjugation we will consider. c

10. Twst conj Twst conj are conjugate, Braid groups Examples are twistedconjugate. but nottwistedconjugate. How to solve the twisted conjugacy problems?

11. Theorem (Elrifai-Morton, 1988): Let u,v2Bn conjugate, where every intermediate conjugate w has Back to the conjugacy problem ElRifai-Morton’s solution Algorithm to solve the conjugacy problem. (ElRifai-Morton, 1988) Compute a finite set, invariant of the conjugacy class. SSS(x) = { conjugates of x, of minimal canonical length} One can compute SSS(x) using the following: Then u and v can be joined through conjugations by simple elements,

12. c1 c2 cr v w1 wr-1 w2 u  Theorem (Elrifai-Morton, 1988): Let u,v2Bn conjugate, Then u and v can be joined through conjugations by simple elements, where every intermediate conjugate w has Back to the conjugacy problem ElRifai-Morton’s solution (can assume positive) c v u c1 c2 cr (left normal form) Each ci is simple

13. Back to the conjugacy problem ElRifai-Morton’s solution Computing SSS(x): x Conjugate by all simple elemets... …keeping elemets of minimal length. If no new element is found, SSS(x) is computed. This solves the conjugacy problem. SSS(x)

14. Twst conj Twisted conjugacy problem Solution First idea: Restrict to positive braids. For every x2Bn, xp is positive for p big enough. Positive! Every braid is twisted conjugate to a positive braid.

15. Twisted conjugacy problem Solution First idea: Restrict to positive braids. Every braid is twisted conjugate to a positive braid. The set of positive twisted-conjugates of x is infinite. But… … … 5 3 3 5 2 (braid) 2 1 1

16. A positive braid x is palindromic-free if it cannot be written as: = 32123 Twisted conjugacy problem Solution Second idea: Restrict to positive, palindromic-free braids. Every positive braid is twisted conjugate to a palindromic-free one. 12321 232 3 1 212

17. Second idea: Restrict to positive, palindromic-free braids. Every positive braid is twisted conjugate to a palindromic-free one. k k Twisted conjugacy problem Solution But… The set of positive, palindromic-free twisted-conjugates of x can be infinite. Example: k k These braids are palindromic-free, for all k. They are twisted conjugate.

18. Positive,MPF(x)= Twisted conjugates of x which are: Palindromic-free of minimal length. Twisted conjugacy problem Solution Third idea: Restrict to positive, palindromic-free braids, of minimal length This is a finite set, invariant of the twisted-conjugacy class. Computing MPF(x), we solve the twisted.conjugacy problem. How to compute it?

19. For usual conjugacy problem… 2 1 2 1 c1 c2 cr 2 1 v w1 wr-1 w2 u  simple simple Simple twisted-conjugation: y x Computing MPF(x) …simple conjugations. For twisted conjugacy problem: 212 121 = 2 1

20. Theorem (Elrifai-Morton, 1988): Let u,v2Bn conjugate, Then u and v can be joined through conjugations by simple elements, where every intermediate conjugate w has Theorem (GM-Ventura, 2008): Let u,v2Bntwisted conjugate, where every intermediate twisted-conjugate w has w1 wr-1 w2  Computing MPF(x) Then u and v can be joined through simple twisted-conjugations, v u All palindromic-free

21. Computing MPF(x) Ingredients of the proof: u v 1 Then use Elifai-Morton’s Theorem.

22. Conclusion H = … f.g.-free f.g.-t.f.-hyperbolic Since AG < Aut (Bn) is orbit decidable, and Bn has solvable twistedconjugacy problem, G has solvable conjugacy problem (decision & search)