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
⟨
x
1
,
x
2
,
...
,
x
n
|
⟨
x
1
,
x
2
⟩
m
1
,
2
=
⟨
x
2
,
x
1
⟩
m
2
,
1
,
...
,
⟨
x
n
-
1
,
x
n
⟩
m
n
-
1
,
n
=
⟨
x
n
,
x
n
-
1
⟩
m
n
,
n
-
1
⟩
{\ 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
m
i
,
j
=
m
j
,
i
∈
{
2
,
3
,
...
,
∞
}
{\ 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
m
<
∞
{\ displaystyle m <\ infty}
⟨
x
i
,
x
j
⟩
m
{\ displaystyle \ langle x_ {i}, x_ {j} \ rangle ^ {m}}
m
{\ displaystyle m}
x
i
{\ displaystyle x_ {i}}
x
j
{\ displaystyle x_ {j}}
x
i
{\ displaystyle x_ {i}}
⟨
x
i
,
x
j
⟩
3
=
x
i
x
j
x
i
{\ displaystyle \ langle x_ {i}, x_ {j} \ rangle ^ {3} = x_ {i} x_ {j} x_ {i}}
or
⟨
x
i
,
x
j
⟩
4th
=
x
i
x
j
x
i
x
j
{\ displaystyle \ langle x_ {i}, x_ {j} \ rangle ^ {4} = x_ {i} x_ {j} x_ {i} x_ {j}}
.
m
i
j
=
∞
{\ displaystyle m_ {ij} = \ infty}
means that there are no relations between and .
x
i
{\ displaystyle x_ {i}}
x
j
{\ displaystyle x_ {j}}
Special cases
Braid groups
Braid groups are included as a special case
m
i
,
i
+
1
=
m
i
+
1
,
i
=
3
∀
i
{\ displaystyle m_ {i, i + 1} = m_ {i + 1, i} = 3 \ \ forall i}
and
m
i
,
j
=
2
∀
∣
i
-
j
∣>
1
{\ displaystyle m_ {i, j} = 2 \ \ forall \ mid ij \ mid> 1}
.
Right-angled Artin groups
Right-angled Artin groups are Artin groups with
m
i
,
j
∈
{
2
,
∞
}
{\ displaystyle m_ {i, j} \ in \ left \ {2, \ infty \ right \}}
for everyone . They play an important role in the 3-dimensional topology.
i
,
j
{\ displaystyle i, j}
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