In themathematical area ofgroup theory,Artin groups, also known asArtin–Tits groups orgeneralized braid groups, are a family of infinite discretegroups defined by simplepresentations. They are closely related withCoxeter groups. Examples arefree groups,free abelian groups,braid groups, and right-angled Artin–Tits groups, among others.

The groups are named afterEmil Artin, due to his early work on braid groups in the 1920s to 1940s,^[1] andJacques Tits who developed the theory of a more general class of groups in the 1960s.^[2]

An Artin–Tits presentation is a grouppresentation${\displaystyle ⟨S\mid R⟩}$ where${\displaystyle S}$ is a (usually finite) set of generators and${\displaystyle R}$ is a set of Artin–Tits relations, namely relations of the form${\displaystyle stst\dots =tsts\dots }$ for distinct${\displaystyle s,t}$ in${\displaystyle S}$, where both sides have equal lengths, and there exists at most one relation for each pair of distinct generators${\displaystyle s,t}$. An Artin–Tits group is a group that admits an Artin–Tits presentation. Likewise, anArtin–Tits monoid is amonoid that, as a monoid, admits an Artin–Tits presentation.

Alternatively, an Artin–Tits group can be specified by the set of generators${\displaystyle S}$ and, for every${\displaystyle s,t}$ in${\displaystyle S}$, the natural number${\displaystyle m_{s,t}⩾2}$ that is the length of the words${\displaystyle stst\dots }$ and${\displaystyle tsts\dots }$ such that${\displaystyle stst\dots =tsts\dots }$ is the relation connecting${\displaystyle s}$ and${\displaystyle t}$, if any. By convention, one puts${\displaystyle m_{s,t}=\mathrm{\infty }}$ when there is no relation${\displaystyle stst\dots =tsts\dots }$ . Formally, if we define${\displaystyle ⟨s,t{⟩}^m}$ to denote an alternating product of${\displaystyle s}$ and${\displaystyle t}$ of length${\displaystyle m}$, beginning with${\displaystyle s}$ — so that${\displaystyle ⟨s,t{⟩}^2=st}$,${\displaystyle ⟨s,t{⟩}^3=sts}$, etc. — the Artin–Tits relations take the form

${\displaystyle ⟨s,t{⟩}^{m_{s,t}}=⟨t,s{⟩}^{m_{t,s}},\text{ where }{m}_{s,t}=m_{t,s}\in \{2,3,\dots ,\mathrm{\infty }\}.}$

The integers${\displaystyle m_{s,t}}$ can be organized into asymmetric matrix, known as theCoxeter matrix of the group.

If${\displaystyle ⟨S\mid R⟩}$ is an Artin–Tits presentation of an Artin–Tits group${\displaystyle A}$, thequotient of${\displaystyle A}$ obtained by adding the relation${\displaystyle s^2=1}$ for each${\displaystyle s}$ of${\displaystyle R}$ is aCoxeter group.Conversely, if${\displaystyle W}$ is a Coxeter group presented by reflections and the relations${\displaystyle s^2=1}$ are removed, the extension thus obtained is an Artin–Tits group. For instance, the Coxeter group associated with the${\displaystyle n}$-strand braid group is thesymmetric group of allpermutations of${\displaystyle \{1,\dots ,n\}}$.

• ${\displaystyle G=⟨S\mid \mathrm{\varnothing }⟩}$ is the free group based on${\displaystyle S}$; here${\displaystyle m_{s,t}=\mathrm{\infty }}$ for all${\displaystyle s,t}$.
• ${\displaystyle G=⟨S\mid \{st=ts\mid s,t\in S\}⟩}$ is the free abelian group based on${\displaystyle S}$; here${\displaystyle m_{s,t}=2}$ for all${\displaystyle s,t}$.
• ${\displaystyle G=⟨{\sigma }_1,\dots ,{\sigma }_{n-1}\mid {\sigma }_i{\sigma }_j{\sigma }_i={\sigma }_j{\sigma }_i{\sigma }_j\text{ for }|i-j|=1,{\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i\text{ for }|i-j|⩾2⟩}$ is the braid group on${\displaystyle n}$ strands; here${\displaystyle m_{{\sigma }_i,{\sigma }_j}=3}$ for${\displaystyle |i-j|=1}$, and${\displaystyle m_{{\sigma }_i,{\sigma }_j}=2}$ for${\displaystyle |i-j|>1}$.

Artin–Tits monoids are eligible forGarside methods based on the investigation of their divisibility relations, and are well understood:

• Artin–Tits monoids are cancellative, and they admit greatest common divisors and conditional least common multiples (a least common multiple exists whenever a common multiple does).
• If${\displaystyle A^{+}}$ is an Artin–Tits monoid, and if${\displaystyle W}$ is the associated Coxeter group, there is a (set-theoretic) section${\displaystyle \sigma }$ of${\displaystyle W}$ into${\displaystyle A^{+}}$, and every element of${\displaystyle A^{+}}$ admits a distinguished decomposition as a sequence of elements in the image of${\displaystyle \sigma }$ ("greedy normal form").

Very few results are known for general Artin–Tits groups. In particular, the following basic questions remainopen in the general case:

– solving theword andconjugacy problems — which areconjectured to be decidable,

– determining torsion — which is conjectured to be trivial,

– determining the center — which is conjectured to be trivial or monogenic in the case when the group is not a direct product ("irreducible case"),

– determining the cohomology — in particular solving the${\displaystyle K(\pi ,1)}$ conjecture, i.e., finding an acyclic complex whosefundamental group is the considered group.

Partial results involving particular subfamilies are gathered below. Among the few known general results, one can mention:

• Artin–Tits groups are infinite countable.
• In an Artin–Tits group${\displaystyle ⟨S\mid R⟩}$, the only relation connecting the squares of the elements${\displaystyle s,t}$ of${\displaystyle S}$ is${\displaystyle s^2{t}^2=t^2{s}^2}$ if${\displaystyle st=ts}$ is in${\displaystyle R}$ (John Crisp and Luis Paris^[3]).
• For every Artin–Tits presentation${\displaystyle ⟨S\mid R⟩}$, the Artin–Tits monoid presented by${\displaystyle ⟨S\mid R⟩}$ embeds in the Artin–Tits group presented by${\displaystyle ⟨S\mid R⟩}$ (Paris^[4]).
• Every (finitely generated) Artin–Tits monoid admits a finite Garside family (Matthew Dyer and Christophe Hohlweg^[5]). As a consequence, the existence of common right-multiples in Artin–Tits monoids is decidable, and reduction of multifractions is effective.

Several important classes of Artin groups can be defined in terms of the properties of the Coxeter matrix.

• An Artin–Tits group is said to be ofspherical type if the associatedCoxeter group${\displaystyle W}$ is finite — the alternative terminology "Artin–Tits group of finite type" is to be avoided, because of its ambiguity: a "finite type group" is just one that admits a finite generating set. Recall that a complete classification is known, the 'irreducible types' being labeled as the infinite series${\displaystyle A_n}$,${\displaystyle B_n}$,${\displaystyle D_n}$,${\displaystyle I_2(n)}$ and six exceptional groups${\displaystyle E_6}$,${\displaystyle E_7}$,${\displaystyle E_8}$,${\displaystyle F_4}$,${\displaystyle H_3}$, and${\displaystyle H_4}$.
• In the case of a spherical Artin–Tits group, the group is a group of fractions for the monoid, making the study much easier. Every above-mentioned problem is solved in the positive for spherical Artin–Tits groups: the word and conjugacy problems are decidable, their torsion is trivial, the center is monogenic in the irreducible case, and thecohomology is determined (Pierre Deligne, by geometrical methods,^[6]Egbert Brieskorn andKyoji Saito, by combinatorial methods^[7]).
• A pure Artin–Tits group of spherical type can be realized as thefundamental group of the complement of a finitehyperplane arrangement in${\displaystyle {\mathbb{C}}^n}$.
• Artin–Tits groups of spherical type arebiautomatic groups (Ruth Charney^[8]).
• In modern terminology, an Artin–Tits group${\displaystyle A}$ is aGarside group, meaning that${\displaystyle A}$ is a group of fractions for the associated monoid${\displaystyle A^{+}}$ and there exists for each element of${\displaystyle A}$ a unique normal form that consists of a finite sequence of (copies of) elements of${\displaystyle W}$ and their inverses ("symmetric greedy normal form")

• An Artin–Tits group is said to beright-angled if all coefficients of the Coxeter matrix are either${\displaystyle 2}$ or${\displaystyle \mathrm{\infty }}$, i.e., all relations are commutation relations${\displaystyle st=ts}$. The names(free) partially commutative group,graph group,trace group,semifree group or evenlocally free group are also common.
• For this class of Artin–Tits groups, a different labeling scheme is commonly used. Anygraph${\displaystyle \mathrm{\Gamma }}$ on${\displaystyle n}$ vertices labeled${\displaystyle 1,2,\dots ,n}$ defines a matrix${\displaystyle M}$, for which${\displaystyle m_{s,t}=2}$ if the vertices${\displaystyle s}$ and${\displaystyle t}$ are connected by an edge in${\displaystyle \mathrm{\Gamma }}$, and${\displaystyle m_{s,t}=\mathrm{\infty }}$ otherwise.
• The class of right-angled Artin–Tits groups includes thefree groups of finite rank, corresponding to a graph with no edges, and the finitely-generatedfree abelian groups, corresponding to acomplete graph. Every right-angled Artin group of rankr can be constructed asHNN extension of a right-angled Artin group of rank${\displaystyle r-1}$, with thefree product anddirect product as the extreme cases. A generalization of this construction is called agraph 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 (theinfinite cyclic group).
• The word and conjugacy problems of a right-angled Artin–Tits group are decidable, the former in linear time, the group is torsion-free, and there is an explicit cellular finite${\displaystyle K(\pi ,1)}$ (John Crisp, Eddy Godelle, and Bert Wiest^[9]).
• Every right-angled Artin–Tits group acts freely and cocompactly on a finite-dimensionalCAT(0) cube complex, its "Salvetti complex". As an application, one can use right-angled Artin groups and their Salvetti complexes to construct groups with givenfiniteness properties (Mladen Bestvina and Noel Brady^[10]) see also (Ian Leary^[11]).

• An Artin–Tits group (and a Coxeter group) is said to be oflarge type if${\displaystyle m_{s,t}⩾3}$ for all generators${\displaystyle s\ne t}$; it is said to be ofextra-large type if${\displaystyle m_{s,t}⩾4}$ for all generators${\displaystyle s\ne t}$.
• Artin–Tits groups of extra-large type are eligible for small cancellation theory. As an application, Artin–Tits groups of extra-large type aretorsion-free and have solvable conjugacy problem (Kenneth Appel and Paul Schupp^[12]).
• Artin–Tits groups of extra-large type are biautomatic (David Peifer^[13]).
• Artin groups of large type are shortlex automatic with regular geodesics (Derek Holt and Sarah Rees^[14]).

Many other families of Artin–Tits groups have been identified and investigated. Here we mention two of them.

• An Artin–Tits group${\displaystyle ⟨S\mid R⟩}$ is said to beof FC type ("flag complex") if, for every subset${\displaystyle S^{\prime }}$ of${\displaystyle S}$ such that${\displaystyle m_{s,t}\ne \mathrm{\infty }}$ for all${\displaystyle s,t}$ in${\displaystyle S^{\prime }}$, the group${\displaystyle ⟨S^{\prime }\mid R\cap S^{\prime }{}^2⟩}$ is of spherical type. Such groups act cocompactly on a CAT(0) cubical complex, and, as a consequence, one can find a rational normal form for their elements and deduce a solution to the word problem (Joe Altobelli and Charney^[15]). An alternative normal form is provided by multifraction reduction, which gives a unique expression by an irreducible multifraction directly extending the expression by an irreducible fraction in the spherical case (Dehornoy^[16]).
• An Artin–Tits group is said to beof affine type if the associated Coxeter group isaffine. They correspond to the extended Dynkin diagrams of the four infinite families${\displaystyle {\tilde{A}}_n}$ for${\displaystyle n⩾1}$,${\displaystyle {\tilde{B}}_n}$,${\displaystyle {\tilde{C}}_n}$ for${\displaystyle n⩾2}$, and${\displaystyle {\tilde{D}}_n}$ for${\displaystyle n⩾3}$, and of the five sporadic types${\displaystyle {\tilde{E}}_6}$,${\displaystyle {\tilde{E}}_7}$,${\displaystyle {\tilde{E}}_8}$,${\displaystyle {\tilde{F}}_4}$, and${\displaystyle {\tilde{G}}_2}$. Affine Artin–Tits groups areof Euclidean type: the associated Coxeter group acts geometrically on a Euclidean space. As a consequence, their center is trivial, and their word problem is decidable (Jon McCammond and Robert Sulway^[17]). In 2019, a proof of the${\displaystyle K(\pi ,1)}$ conjecture was announced for all affine Artin–Tits groups (Mario Salvetti and Giovanni Paolini^[18]).

• Free partially commutative monoid
• Artinian group (an unrelated notion)
• Non-commutative cryptography
• Elementary abelian group

• Charney, Ruth (2007), "An introduction to right-angled Artin groups",Geometriae Dedicata,125 (1):141–158,arXiv:math/0610668,doi:10.1007/s10711-007-9148-6,MR 2322545
• Godelle, Eddy; Paris, Luis (2012),Basic questions on Artin–Tits groups, CRM Series, vol. 14, Ed. Norm., Pisa, pp. 299–311,arXiv:1105.1048,doi:10.1007/978-88-7642-431-1_13,ISBN 978-88-7642-430-4,MR 3203644
• McCammond, Jon (2017), "The mysterious geometry of Artin groups",Winter Braids Lecture Notes,4 (Winter Braids VII (Caen, 2017)):1–30,doi:10.5802/wbln.17,MR 3922033
• Flores, Ramon;Kahrobaei, Delaram; Koberda, Thomas (2019). "Algorithmic problems in right-angled Artin groups: complexity and applications".Journal of Algebra.519:111–129.arXiv:1802.04870.doi:10.1016/j.jalgebra.2018.10.023.MR 3874519.

• Braid groups

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