Artin–Tits group

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 ${\displaystyle \{1,\ldots ,n\}}$.

• ${\displaystyle G=\langle S\mid \emptyset \rangle }$ 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 ${\displaystyle \vert i-j\vert >1}$.

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.

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

• 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, 14, Ed. Norm., Pisa, pp. 299–311, 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.