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