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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.
A group G is said to be left-orderable if it admits a total order...
Orderings Left-orderable groups
G
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 cG
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)
Introduction Bi-orderable groups
Left-orderable groups
Bi-orderable groups
No torsion
R integral domain ) RG integral domain
No generalized torsion
Unicity of roots
A, C bi-orderable
Orderings Group extensions
A, C left-orderable ) B left-orderableA, 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.
Examples
(lexicographical order)
Fn is bi-orderable.
is bi-orderable.
Magnus expansionMagnus expansion
(non-commutative variables)
Order in : grlex on the monomials
is injective.
Order in Fn:
Free abelian and free
lex on the series.
Examples
Thompson’s F is bi-orderableThompson’s F is bi-orderable
(Brin-Squier, 1985)
f 2 F is positive if its leftmost slope 1 is >1.f 2 F is positive if its leftmost slope 1 is >1.
Thompson’s F
Examples Braid groups
Braid groups are left-orderableBraid groups are left-orderable (Dehornoy, 1994)
Braids in Bn can be seen as automorphisms of the n-times puncturted disc
Examples Braid groups
Braid groups are left-orderableBraid groups are left-orderable (Dehornoy, 1994)
Examples Braid groups
Braid groups are left-orderableBraid 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.
=
Examples Braid groups
Braid groups Bn are not bi-orderable for n>2Braid groups Bn are not bi-orderable for n>2
Roots are not unique.
Examples Braid groups
Pure braid groups are bi-orderablePure 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.The lex order is a bi-order.
Braided Thompson’s groups Definition
T-
T+
Element of Thompson’s V (with n leaves)
Braided Thompson’s groups Definition
T-
T+
Element of Thompson’s V (with n leaves)
1
2
3 45
51 4
3 2
Element of Bn.
Braided Thompson’s groups Definition
Element of Thompson’s V (with n leaves) Element of Bn.
Element of BVElement of BV
T-
T+
b
Brin (2004)
Dehornoy (2004)
Braided Thompson’s groups Definition
Elements of BV admit distinct representations:
Adding carets&
doubling strings
=
Braided Thompson’s groups Definition
Multiplication in BV:
= =Same tree
Braided Thompson’s groups Subgroups
BF ½ BVBF ½ BV Elements of BF:
Pure braid
Braided Thompson’s groups Subgroups
From the morphisms
BF ½ BVBF ½ BV Elements of BF:
Pure braid
PBV ½ BVPBV ½ BV
we obtain a morphism
Braided Thompson’s groups Subgroups
Elements of PBV:
Pure braid Same tree
Notice that:
Ordering braided Thompson’s groups BV and BF
Recall that: Bn is left-orderableBn is left-orderable Pn is bi-orderablePn is bi-orderable
Now: BV is left-orderableBV is left-orderable
BV cannot be bi-orderable, since it contains Bn.
Theorem: (Burillo-GM, 2006) BF is bi-orderableTheorem: (Burillo-GM, 2006) BF is bi-orderable
Proof: We will order PBV.
(Dehornoy, 2005)
Ordering braided Thompson’s groups PBV
PBV contains many copies of the pure braid group Pn.
Fixing a tree T :
Each copy of Pn overlaps with several copies of Pn+1.Adding carets
&doubling strings
Doubling the i-th string
Ordering braided Thompson’s groups PBV
T
T
is a directed system. 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.Lemma: If a pure braid is positive, and we double a string, the result is positive.
Ordering braided Thompson’s groups Conclusion
Lemma: If a pure braid is positive, and we double a string, the result is positive.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: PBV is bi-orderable.
Corollary: BF is bi-orderable.Corollary: BF is bi-orderable.
( in F )
( in Pn )
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