# Knot at Lunch, November 7, 2007

## Invitation

Dear Knot at Lunch People,

We will have our next fall lunch on Wednesday November 7, at 12 noon, not in our usual place, but instead at the department "meeting room", Bahen 6290B (the little room by the mail room, entrance from the administrative offices)

As always, please bring brown-bag lunch and fresh ideas. The agenda: I will lead an informal discussion of "Expansions for Groups".

As always, if you know anyone I should add to this mailing list or if you wish to be removed from this mailing list please let me know. To prevent junk accumulation in mailboxes, I will actively remove inactive people unless they request otherwise.

Best,

Dror.

## The Agenda

• Definition of "an expansion".
• Solving equations in groups.
• Examples: ${\displaystyle {\mathbb {Z} }}$, ${\displaystyle {\mathbb {Z} }^{n}}$, ${\displaystyle F_{n}}$, ${\displaystyle PB_{n}}$, and ${\displaystyle B_{n}}$.
• Relative expansions.
• Functorial expansions.
• Expansions for other algebraic structures.

### Note

If ${\displaystyle \sigma _{j}}$ denote the standard generators of the (non-pure) braid group ${\displaystyle B_{n}}$ then the pure braid group ${\displaystyle PB_{n}}$ is generated by elements ${\displaystyle x_{ij}}$ with ${\displaystyle i and

${\displaystyle x_{ij}=(\sigma _{j-2}\cdots \sigma _{i})^{-1}\sigma _{j-2}^{2}(\sigma _{j-2}\cdots \sigma _{i})}$

with relations (always ${\displaystyle i):

${\displaystyle [a_{ijk},x_{ij}]=[a_{ijk},x_{ik}]=[a_{ijk},x_{jk}]=1}$ where ${\displaystyle a_{ijk}:=x_{ij}x_{ik}x_{jk}}$

and

${\displaystyle [x_{ij},x_{kl}]=[x_{il},x_{jk}]=[x_{ik},x_{ij}^{-1}x_{jl}x_{ij}]=1}$.