Artin group

In the mathematical area of group theory, Artin groups, also known as Artin–Tits groups or generalized braid groups, are a family of infinite discrete groups defined by simple presentations. They are closely related with Coxeter groups. Examples are free groups, free abelian groups, braid groups, and right-angled Artin–Tits groups, among others. In the mathematical area of group theory, Artin groups, also known as Artin–Tits groups or generalized braid groups, are a family of infinite discrete groups defined by simple presentations. They are closely related with Coxeter groups. Examples are free groups, free abelian groups, braid groups, and right-angled Artin–Tits groups, among others. The groups are named after Emil Artin, due to his early work on braid groups in the 1920s to 1940s, and Jacques Tits who developed the theory of a more general class of groups in the 1960s. An Artin–Tits presentation is a group presentation ⟨ S ∣ R ⟩ {displaystyle langle Smid R angle } where S {displaystyle S} is a (usually finite) set of generators and R {displaystyle R} is a set of Artin–Tits relations, namely relations of the form s t s t … = t s t s … {displaystyle ststldots =tstsldots } for distinct s , t {displaystyle s,t} in S {displaystyle S} , where both sides have equal lengths, and there exists at most one relation for each pair of distinct generators s , t {displaystyle s,t} . An Artin–Tits group is a group that admits an Artin–Tits presentation. Likewise, an Artin–Tits monoid is a monoid that, as a monoid, admits an Artin–Tits presentation. Alternatively, an Artin–Tits group can be specified by the set of generators S {displaystyle S} and, for every s , t {displaystyle s,t} in S {displaystyle S} , the natural number m s , t ⩾ 2 {displaystyle m_{s,t}geqslant 2} that is the length of the words s t s t … {displaystyle ststldots } and t s t s … {displaystyle tstsldots } such that s t s t … = t s t s … {displaystyle ststldots =tstsldots } is the relation connecting s {displaystyle s} and t {displaystyle t} , if any. By convention, one puts m s , t = ∞ {displaystyle m_{s,t}=infty } when there is no relation s t s t … = t s t s … {displaystyle ststldots =tstsldots } . Formally, if we define ⟨ s , t ⟩ m {displaystyle langle s,t angle ^{m}} to denotes an alternating product of s {displaystyle s} and t {displaystyle t} of length m {displaystyle m} , beginning with s {displaystyle s} — so that ⟨ s , t ⟩ 2 = s t {displaystyle langle s,t angle ^{2}=st} , ⟨ s , t ⟩ 3 = s t s {displaystyle langle s,t angle ^{3}=sts} , etc. — the Artin–Tits relations take the form The integers m s , t {displaystyle m_{s,t}} can be organized into a symmetric matrix, known as the Coxeter matrix of the group. If ⟨ S ∣ R ⟩ {displaystyle langle Smid R angle } is an Artin–Tits presentation of a Artin–Tits group A {displaystyle A} , the quotient of A {displaystyle A} obtained by adding the relation s 2 = 1 {displaystyle s^{2}=1} for each s {displaystyle s} of R {displaystyle R} is a Coxeter group. Conversely, if W {displaystyle W} is a Coxeter group presented by reflections and the relations s 2 = 1 {displaystyle s^{2}=1} are removed, the extension thus obtained is an Artin–Tits group. For instance, the Coxeter group associated with the n {displaystyle n} -strand braid group is the symmetric group of all permutations of { 1 , … , n } {displaystyle {1,ldots ,n}} . Artin–Tits monoids are eligible for Garside methods based on the investigation of their divisibility relations, and are well understood: