# Artin–Tits 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.

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

## Definition

An Artin–Tits presentation is a group presentation ${\displaystyle \langle S\mid R\rangle }$ 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 ${\displaystyle stst\ldots =tsts\ldots }$ 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, 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 ${\displaystyle S}$ and, for every ${\displaystyle s,t}$ in ${\displaystyle S}$, the natural number ${\displaystyle m_{s,t}\geqslant 2}$ that is the length of the words ${\displaystyle stst\ldots }$ and ${\displaystyle tsts\ldots }$ such that ${\displaystyle stst\ldots =tsts\ldots }$ is the relation connecting ${\displaystyle s}$ and ${\displaystyle t}$, if any. By convention, one puts ${\displaystyle m_{s,t}=\infty }$ when there is no relation ${\displaystyle stst\ldots =tsts\ldots }$ . Formally, if we define ${\displaystyle \langle s,t\rangle ^{m}}$ to denotes an alternating product of ${\displaystyle s}$ and ${\displaystyle t}$ of length ${\displaystyle m}$, beginning with ${\displaystyle s}$ — so that ${\displaystyle \langle s,t\rangle ^{2}=st}$, ${\displaystyle \langle s,t\rangle ^{3}=sts}$, etc. — the Artin–Tits relations take the form

${\displaystyle \langle s,t\rangle ^{m_{s,t}}=\langle t,s\rangle ^{m_{t,s}},{\text{ where }}m_{s,t}=m_{t,s}\in \{2,3,\ldots ,\infty \}.}$

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

If ${\displaystyle \langle S\mid R\rangle }$ is an Artin–Tits presentation of an Artin–Tits group ${\displaystyle A}$, the quotient of ${\displaystyle A}$ obtained by adding the relation ${\displaystyle s^{2}=1}$ for each ${\displaystyle s}$ of ${\displaystyle R}$ is a Coxeter 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 the symmetric group of all permutations of ${\displaystyle \{1,\ldots ,n\}}$.

## Examples

• ${\displaystyle G=\langle S\mid \emptyset \rangle }$ is the free group based on ${\displaystyle S}$; here ${\displaystyle m_{s,t}=\infty }$ for all ${\displaystyle s,t}$.
• ${\displaystyle G=\langle S\mid \{st=ts\mid s,t\in S\}\rangle }$ is the free abelian group based on ${\displaystyle S}$; here ${\displaystyle m_{s,t}=2}$ for all ${\displaystyle s,t}$.
• ${\displaystyle G=\langle \sigma _{1},\ldots ,\sigma _{n-1}\mid \sigma _{i}\sigma _{j}\sigma _{i}=\sigma _{j}\sigma _{i}\sigma _{j}{\text{ for }}\vert i-j\vert =1,\sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}{\text{ for }}\vert i-j\vert \geqslant 2\rangle }$ is the braid group on ${\displaystyle n}$ strands; here ${\displaystyle m_{\sigma _{i},\sigma _{j}}=3}$ for ${\displaystyle \vert i-j\vert =1}$, and ${\displaystyle m_{\sigma _{i},\sigma _{j}}=2}$ for ${\displaystyle \vert i-j\vert >1}$.

## General properties

Artin–Tits monoids are eligible for Garside 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 remain open in the general case:

– solving the word and conjugacy problems — which are conjectured 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 whose fundamental 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 ${\displaystyle \langle S\mid R\rangle }$, 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 ${\displaystyle \langle S\mid R\rangle }$, the Artin–Tits monoid presented by ${\displaystyle \langle S\mid R\rangle }$ embeds in the Artin–Tits group presented by ${\displaystyle \langle S\mid R\rangle }$ (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.

## Particular classes of Artin–Tits groups

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

### Artin–Tits groups of spherical type

• An Artin–Tits group is said to be of spherical type if the associated Coxeter group ${\displaystyle W}$ is finite — the alternative terminology "Artin–Tits group of finite type" is to avoid, 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 the cohomology is determined (Pierre Deligne, by geometrical methods,[6] Egbert Brieskorn and Kyoji Saito, by combinatorial methods [7]).
• A pure Artin–Tits group of spherical type can be realized as the fundamental group of the complement of a finite hyperplane arrangement in ${\displaystyle \mathbb {C} ^{n}}$.
• Artin–Tits groups of spherical type are biautomatic groups (Ruth Charney[8]).
• In modern terminology, an Artin–Tits group ${\displaystyle A}$ is a Garside 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")

### Right-angled Artin groups

• An Artin–Tits group is said to be right-angled if all coefficients of the Coxeter matrix are either ${\displaystyle 2}$ or ${\displaystyle \infty }$, i.e., all relations are commutation relations ${\displaystyle st=ts}$. The names (free) partially commutative group, graph group, trace group, semifree group or even locally free group are also common.
• For this class of Artin–Tits groups, a different labeling scheme is commonly used. Any graph ${\displaystyle \Gamma }$ on ${\displaystyle n}$ vertices labeled ${\displaystyle 1,2,\ldots ,n}$ 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 ${\displaystyle \Gamma }$, and ${\displaystyle m_{s,t}=\infty }$ otherwise.
• The class of right-angled Artin–Tits groups includes the free groups of finite rank, corresponding to a graph with no edges, and the finitely-generated free abelian groups, corresponding to a complete graph. Every right-angled Artin group of rank r can be constructed as HNN extension of a right-angled Artin group of rank ${\displaystyle r-1}$, with the free product and 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).
• 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-dimensional CAT(0) cube complex, its "Salvetti complex". As an application, one can use right-angled Artin groups and their Salvetti complexes to construct groups with given finiteness properties (Mladen Bestvina and Joel Brady [10]).

### Artin–Tits groups of large type

• An Artin–Tits group (and a Coxeter group) is said to be of large type if ${\displaystyle m_{s,t}\geqslant 3}$ for all generators ${\displaystyle s\neq t}$; it is said to be of extra-large type if ${\displaystyle m_{s,t}\geqslant 4}$ for all generators ${\displaystyle s\neq t}$.
• Artin–Tits groups of extra-large type are eligible for small cancellation theory. As an application, Artin–Tits groups of extra-large type are torsion-free and have solvable conjugacy problem (Kenneth Appel and Paul Schupp[11]).
• Artin–Tits groups of extra-large type are biautomatic (David Peifer[12]).
• Artin groups of large type are shortlex automatic with regular geodesics (Derek Holt and Sarah Rees[13]).

### Other types

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

• An Artin–Tits group ${\displaystyle \langle S\mid R\rangle }$ is said to be of FC type ("flag complex") if, for every subset ${\displaystyle S'}$ of ${\displaystyle S}$ such that ${\displaystyle m_{s,t}\neq \infty }$ for all ${\displaystyle s,t}$ in ${\displaystyle S'}$, the group ${\displaystyle \langle S'\mid R\cap S'{}^{2}\rangle }$ 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 [14]). 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[15]).
• An Artin–Tits group is said to be of affine type if the associated Coxeter group is affine. They correspond to the extended Dynkin diagrams of the four infinite families ${\displaystyle {\widetilde {A}}_{n}}$ for ${\displaystyle n\geqslant 1}$, ${\displaystyle {\widetilde {B}}_{n}}$, ${\displaystyle {\widetilde {C}}_{n}}$ for ${\displaystyle n\geqslant 2}$, and ${\displaystyle {\widetilde {D}}_{n}}$ for ${\displaystyle n\geqslant 3}$, and of the five sporadic types ${\displaystyle {\widetilde {E}}_{6}}$, ${\displaystyle {\widetilde {E}}_{7}}$, ${\displaystyle {\widetilde {E}}_{8}}$, ${\displaystyle {\widetilde {F}}_{4}}$, et ${\displaystyle {\widetilde {G}}_{2}}$. Affine Artin–Tits groups are of 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 [16]). Very recently, a proof of the ${\displaystyle K(\pi ,1)}$ conjecture was announced for all affine Artin–Tits groups (Mario Salvetti and Giovanni Paolini[17]).

## References

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