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changed Wikilink to Secular_variations_of_the_planetary_orbits - that article was mis-named

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In [[mathematics]], an '''Artin group''' (or '''generalized braid group''') is a [[group (mathematics)|group]] with a [[presentation of a group|presentation]] of the form
:<math>\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</math>
where
:<math>m_{i,j} = m_{j,i} \in \{2,3,\ldots, \infty\}</math>.

For <math>m < \infty</math>, <math>\langle x_i, x_j \rangle^m</math> denotes an alternating product of <math>x_i</math> and <math>x_j</math> of length <math>m</math>, beginning with <math>x_i</math>. For example,

:<math>\langle x_i, x_j \rangle^3 = x_ix_jx_i</math>

and

:<math>\langle x_i, x_j \rangle^4 = x_ix_jx_ix_j</math>.

If <math>m=\infty</math>, then there is (by convention) no relation for <math>x_i</math> and <math>x_j</math>.

The integers <math>m_{i,j}</math> can be organized into a [[symmetric matrix]], known as the [[Coxeter matrix]] of the group. Each Artin group has as a quotient the [[Coxeter group]] with the same set of generators and Coxeter matrix. The [[kernel (set theory)|kernel]] of the [[homomorphism]] to the associated Coxeter group, known as the '''pure Artin group''', is generated by relations of the form <math>{x_i}^2=1</math>.

== Classes of Artin groups ==

[[Braid group]]s are examples of Artin groups, with Coxeter matrix <math>m_{i,i+1} = 3</math> and <math>m_{i,j}=2</math> for <math>|i-j|>1.</math> Several important classes of Artin groups can be defined in terms of the properties of the Coxeter matrix.

=== Artin groups of finite type ===

If ''M'' is a Coxeter matrix of finite type, so that the corresponding [[Coxeter group]] ''W'' = ''A''(''M'') is finite, then the Artin group ''A'' = ''A''(''M'') is called an '''Artin group of finite type'''. The 'irreducible types' are labeled as ''A''''n'' , ''B''''n'' = ''C''''n'' , ''D''''n'' , ''I''''2''(''n'') , ''F''''4'' , ''E''''6'' , ''E''''7'' , ''E''''8'' , ''H''''3'' , ''H''''4'' .
A pure Artin group of finite type can be realized as the [[fundamental group]] of the complement of a finite [[hyperplane arrangement]] in '''C'''''n''. [[Pierre Deligne]] and Brieskorn-Saito have used this geometric description to compute the center of ''A'', its [[group cohomology|cohomology]], and to solve the [[word problem for groups|word]] and [[conjugacy problem|conjugacy]] problems.

=== Right-angled Artin groups ===

If ''M'' is a matrix all of whose elements are equal to 2 or ∞, then the corresponding Artin group is called a '''right-angled Artin group''', but also a '''(free) partially commutative group''', '''graph group''', '''trace group''', '''semifree group''' or even '''locally free group'''. For this class of Artin groups, a different labeling scheme is commonly used. Any [[graph (mathematics)|graph]] '''Γ''' on ''n'' vertices labeled 1, 2, …, n defines a matrix ''M'', for which ''m''''ij'' = 2 if ''i'' and ''j'' are connected by an edge in '''Γ''', and ''m''''ij'' = ∞ otherwise. The right-angled Artin group ''A''('''Γ''') associated with the matrix ''M'' has ''n'' generators ''x''1, ''x''2, …, ''x''n and relations
: <math> x_i x_j = x_j x_i \quad </math> whenever ''i'' and ''j'' are connected by an edge in <math>\Gamma.</math>

The class of right-angled Artin groups includes the [[free group]]s of finite rank, corresponding to a graph with no edges, and the finitely-generated [[free abelian group]]s, corresponding to a [[complete graph]]. In fact, every right-angled Artin group of rank ''r'' can be constructed as [[HNN extension]] of a right-angled Artin group of rank ''r''-1, with the [[free product]] and [[Direct product of groups|direct product]] as the extreme cases. A generalization of this construction is called a [[graph product of groups]]. A right-angled Artin group is a special case of this product, with every vertex/operand of the graph-product being a free group of rank one (the [[infinite cyclic group]]).

Mladen Bestvina and Noel Brady constructed a nonpositively curved cubical complex ''K'' whose fundamental group is a given right-angled Artin group ''A''('''Γ'''). They applied [[Morse theory|Morse-theoretic]] arguments to their geometric description of Artin groups and exhibited first known examples of groups with the property (FP2) that are not [[finitely presented group|finitely presented]].

== Other Artin Groups ==

We define that an Artin group or a [[Coxeter group]] is of ''large type'' if ''m''i j ≥ 3 for all ''i ≠ j''. We say that an Artin group or a [[Coxeter group]] is of ''extra-large'' type if ''m''i j ≥ 4 for all ''i ≠ j''.

[[Kenneth Appel]] and P.E. Schupp looked further into Artin groups and the properties that hold true for them. They proved four theorems, which were known to be true for [[Coxeter group]]s, and showed that they also held for Artin groups. Appel and Schupp had discovered that they could study extra-large Artin and [[Coxeter group]]s through the techniques of small cancellation theory. They also discovered that they could use a "refinement" of these same techniques to work with these groups of large type.<ref name="Artin">Appel, Kenneth I., and P. E. Schupp. "Artin Groups and Infinite Coxeter Groups." Inventiones Mathematicae 72.2 (1983): 201-220</ref>

'''Theorem 1''' : Let ''G'' be an Artin or Coxeter group of extra-large type. If ''J ⊆ I'' then ''G''J has a presentation defined by the Coxeter matrix ''M''J and the generalized word problem for ''G''J in ''G'' is solvable. If ''J, K'' ⊆ ''I'' then ''G''J ∩ ''G''K = ''G'' (J ∩ K).

'''Theorem 2''' : An Artin group of extra-large type is torsion-free.

'''Theorem 3''' : Let ''G'' be an Artin group of extra-large type. Then the set {''a''i2 : ''i ∈ I''} freely generates a free subgroup of ''G''.

'''Theorem 4''' : An Artin or Coxeter group of extra-large type has solvable conjugacy problem.

== See also ==
* [[Free partially commutative monoid]]
* [[Artinian group]] (an unrelated notion)

== References ==
<references />
* [[Mladen Bestvina]], Noel Brady, ''Morse theory and finiteness properties of groups''. Invent. Math. 129 (1997), no. 3, 445-470.
* [[Pierre Deligne]], ''Les immeubles des groupes de tresses généralisés''. Invent. Math. 17 (1972), 273-302.
* [[Egbert Brieskorn]], Kyoji Saito, ''Artin-Gruppen und Coxeter-Gruppen''. Invent. Math. 17 (1972), 245--271.
* RUTH CHARNEY, [http://people.brandeis.edu/~charney/papers/RAAGfinal.pdf AN INTRODUCTION TO RIGHT-ANGLED ARTIN GROUPS]
* MONTSERRAT CASALS-RUIZ AND ILYA V. KAZACHKOV, [http://arxiv.org/pdf/0810.4867v2.pdf ON SYSTEMS OF EQUATIONS OVER FREE PARTIALLY COMMUTATIVE GROUPS]
* EVGENII S. ESYP, ILYA V. KAZACHKOV, AND VLADIMIR N. REMESLENNIKOV, [http://arxiv.org/pdf/math/0512401.pdf DIVISIBILITY THEORY AND COMPLEXITY OF ALGORITHMS FOR FREE PARTIALLY COMMUTATIVE GROUPS]
* Susan Hermiller, John Meier, [http://www.math.unl.edu/~shermiller2/webppr/graphproduct.pdf Algorithms and geometry for graph products of groups]

[[Category:Group theory]]
[[Category:Geometric group theory]]
[[Category:Braid groups]]
[[Category:Properties of groups]]

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en>Tfr000: changed Wikilink to Secular_variations_of_the_planetary_orbits - that article was mis-named