Artin group

In mathematics , an Artin group is a term from the mathematical branch of group theory . This particular class of group is named after the mathematician Emil Artin .

definition

An Artin group is a group with a presentation of the form

${\displaystyle {\Big \langle }x_{1},x_{2},\ldots ,x_{n}{\Big |}\langle x_{1},x_{2}\rangle ^{m_{1,2}}=\langle x_{2},x_{1}\rangle ^{m_{2,1}},\ldots ,\langle x_{n-1},x_{n}\rangle ^{m_{n-1,n}}=\langle x_{n},x_{n-1}\rangle ^{m_{n,n-1}}{\Big \rangle }}$

With

${\displaystyle m_{i,j}=m_{j,i}\in \{2,3,\ldots ,\infty \}}$.

For means the alternating product of the length of and , starting with . So for example ${\displaystyle m<\infty }$ ${\displaystyle \langle x_{i},x_{j}\rangle ^{m}}$ ${\displaystyle m}$ ${\displaystyle x_{i}}$ ${\displaystyle x_{j}}$ ${\displaystyle x_{i}}$

${\displaystyle \langle x_{i},x_{j}\rangle ^{3}=x_{i}x_{j}x_{i}}$

or

${\displaystyle \langle x_{i},x_{j}\rangle ^{4}=x_{i}x_{j}x_{i}x_{j}}$.

${\displaystyle m_{ij}=\infty }$means that there are no relations between and . ${\displaystyle x_{i}}$ ${\displaystyle x_{j}}$

Special cases

Braid groups

→ Main article : Braid group

Braid groups are included as a special case

${\displaystyle m_{i,i+1}=m_{i+1,i}=3\ \forall i}$

and

${\displaystyle m_{i,j}=2\ \forall \mid i-j\mid >1}$.

Right-angled Artin groups

Right-angled Artin groups are Artin groups with

${\displaystyle m_{i,j}\in \left\{2,\infty \right\}}$

for everyone . They play an important role in the 3-dimensional topology. ${\displaystyle i,j}$