Bi-orderings on pure braided Thompson's groups

1. Bi-orderings on pure braided Thompson's groups Juan González-Meneses Universidad de Sevilla Joint with José Burillo. Les groupes de Thompson: nouveaux développements et interfaces CIRM. Luminy, June 2008.

2. Orderings Left-orderable groups A group G is said to be left-orderable if it admits a total order... G

3. a < b Orderings Left-orderable groups A group G is said to be left-orderable if it admits a total order... … invariant under left-multiplication. G cc

4. Orderings Bi-orderable groups A group G is said to be left-orderable if it admits a total order... … invariant under left-multiplication. A group G is said to be bi-orderable if it admits a total order... … invariant under left & right-multiplication. (In particular, every inner automorphism preserves the order)

5. No torsion Left-orderable groups R integral domain ) RG integral domain No generalized torsion Bi-orderable groups Unicity of roots Introduction Bi-orderable groups

6. A, C bi-orderable )B bi-orderable Lexicographical order in Cn A. Orderings Group extensions A, C left-orderable )B left-orderable B = Cn A The action of C on A preserves <

7. is bi-orderable. is injective. Order in : Examples Free abelian and free (lexicographical order) Fn is bi-orderable. Magnus expansion (non-commutative variables)  lex on the series. grlex on the monomials Order in Fn:

8. Examples Thompson’s F Thompson’s F is bi-orderable (Brin-Squier, 1985) f2F is positiveif its leftmost slope 1 is >1.

9. Examples Braid groups (Dehornoy, 1994) Braid groups are left-orderable

10. Examples Braid groups (Dehornoy, 1994) Braid groups are left-orderable Braids in Bn can be seen as automorphisms of the n-times puncturted disc

11. Examples Braid groups (Dehornoy, 1994) Braid groups are left-orderable (Fenn, Greene, Rolfsen, Rourke, Wiest, 1999) A braid is positive if the leftmost non-horizontal curve in the image of the diameter goes up.

12. = Examples Braid groups Braid groups Bn are not bi-orderable for n>2 Roots are not unique.

13. The lex order is a bi-order. Examples Braid groups Pure braid groups are bi-orderable (Rolfsen-Zhu, 1997) (Kim-Rolfsen, 2003) Pure braids can be combed. Each Fk admits a Magnus ordering. The actions respect these orderings.

14. Braided Thompson’s groups Definition Element of Thompson’s V (with n leaves) T+ T-

15. 1 2 5 3 4 1 4 3 2 5 Braided Thompson’s groups Definition Element of Thompson’s V (with n leaves) Element of Bn. T- T+

16. Braided Thompson’s groups Definition Element of Thompson’s V (with n leaves) Element of Bn. T- Element of BV b Brin (2004) Dehornoy (2004) T+

17. Braided Thompson’s groups Definition Elements of BV admit distinct representations: Adding carets & doubling strings =

18. = = Braided Thompson’s groups Definition Multiplication in BV: Same tree

19. Pure braid Braided Thompson’s groups Subgroups BF ½ BV Elements of BF:

20. From the morphisms we obtain a morphism Braided Thompson’s groups Subgroups BF ½ BV Elements of BF: Pure braid PBV ½ BV

21. Same tree Pure braid Braided Thompson’s groups Subgroups Elements of PBV: Notice that:

22. Ordering braided Thompson’s groups BV and BF Recall that: Bn is left-orderable Pn is bi-orderable BV is left-orderable (Dehornoy, 2005) Now: BV cannot be bi-orderable, since it contains Bn. Theorem:(Burillo-GM, 2006)BF is bi-orderable Proof: We will order PBV.

23. Adding carets & doubling strings Ordering braided Thompson’s groups PBV PBV contains many copies of the pure braid group Pn. Fixing a tree T : Each copy of Pnoverlaps with several copies of Pn+1. Doubling the i-th string

24. Ordering braided Thompson’s groups PBV T T

25. is a directed system. Ordering braided Thompson’s groups PBV Each copy Pn,T of Pn is bi-ordered: Are these orderings compatible with the direct limit? Lemma: If a pure braid is positive, and we double a string, the result is positive.

26. Ordering braided Thompson’s groups Conclusion Lemma: If a pure braid is positive, and we double a string, the result is positive. Proof: Study in detail how doubling a string affects the combing. Corollary: PBV is bi-orderable. Corollary: BF is bi-orderable. ( in F ) ( in Pn )

27. J. Burillo, J. González-Meneses. Bi-orderings on pure braided Thompson's groups. Quarterly J. of Math. 59 (1), 2008, 1-14. arxiv.org/abs/math/0608646