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

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

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

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A group G is said to be left-orderable if it admits a total order... Orderings Bi-orderable groups … 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)

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

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

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

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Examples Thompsons F is bi-orderable (Brin-Squier, 1985) f 2 F is positive if its leftmost slope 1 is >1. Thompsons F

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

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

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

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

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

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Braided Thompsons groups Definition T-T- T+T+ Element of Thompsons V (with n leaves)

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Braided Thompsons groups Definition T-T- T+T+ Element of Thompsons V (with n leaves) 1 2 3 4 5 5 1 4 3 2 Element of B n.

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Braided Thompsons groups Definition Element of Thompsons V (with n leaves)Element of B n. Element of BV T-T- T+T+ b Brin (2004) Dehornoy (2004)

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Braided Thompsons groups Definition Elements of BV admit distinct representations: Adding carets & doubling strings =

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Braided Thompsons groups Definition Multiplication in BV: == Same tree

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Braided Thompsons groups Subgroups BF ½ BV Elements of BF: Pure braid

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Braided Thompsons groups Subgroups From the morphisms BF ½ BV Elements of BF: Pure braid PBV ½ BV we obtain a morphism

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Braided Thompsons groups Subgroups Elements of PBV: Pure braid Same tree Notice that:

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Ordering braided Thompsons groups BV and BF Recall that: B n is left-orderable P n is bi-orderable Now: BV is left-orderable BV cannot be bi-orderable, since it contains B n. Theorem: (Burillo-GM, 2006) BF is bi-orderable Proof: We will order PBV. (Dehornoy, 2005)

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Ordering braided Thompsons groups PBV PBV contains many copies of the pure braid group P n. Fixing a tree T : Each copy of P n overlaps with several copies of P n + 1. Adding carets & doubling strings Doubling the i -th string

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Ordering braided Thompsons groups PBV T T

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is a directed system. Ordering braided Thompsons groups PBV Each copy P n,T of P n 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.

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Ordering braided Thompsons 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 P n )

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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

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