Artin monoid and Artin group presentations, and the canonical monoid-to-group map

Let $S$ be a finite set and let $m$ be a Coxeter matrix on $S$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), so that $m(s,s)=1$ and $m(s,t)=m(t,s)\in \{2,3,\dots \}\cup \{\mathrm{\infty }\}$ for $s\mathrm{\ne }t$.

(1) Words. Let $S^{\ast }$ be the set of finite words $u=s_1{s}_2\cdots s_k$ in the alphabet $S$ ($k\ge 0$, letters read from left to right, concatenation written by juxtaposition, empty word $\epsilon$; Words in an alphabet with formal inverses, elementary cancellation, and reduced words). Concatenation makes $S^{\ast }$ a monoid with identity $\epsilon$ (Semigroup and monoid).

(2) Braid pairs. Let $s\mathrm{\ne }t$ in $S$ with $m:=m(s,t)<\mathrm{\infty }$. The braid words of length $m$ between $s$ and $t$ are the strictly alternating words $u=u_1{u}_2\cdots u_m$ and $v=v_1{v}_2\cdots v_m$ defined by $u_i=s$ for odd $i$ and $u_i=t$ for even $i$, while $v_i=t$ for odd $i$ and $v_i=s$ for even $i$. A braid pair is an unordered pair $\{u,v\}$ arising this way; $m=2$ contributes the commuting pair $\{st,ts\}$ and $m=\mathrm{\infty }$ contributes nothing.

(3) The Artin monoid. A congruence on $S^{\ast }$ is an equivalence relation $\sim$ on $S^{\ast }$ (Equivalence relation, equivalence class, and the quotient set $A\mathrm{/}\sim$) with $u\sim v\Rightarrow xuy\sim xvy$ for all $x,y\in S^{\ast }$. The total relation is a congruence and the intersection of a nonempty family of congruences is a congruence, so there is a smallest congruence ${\equiv }^{+}$ containing every braid pair: take the intersection of all congruences containing the braid pairs. Reflexivity, symmetry, transitivity and compatibility with concatenation hold in the intersection because they hold in each of its members. Put

$A^{+}:=A^{+}(S,m):=S^{\ast }\mathrm{/}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\equiv }^{+},$

write $[u]$ for the class of a word $u$, and define $[u]\cdot [v]:=[uv]$. This multiplication is well defined: if $u{\equiv }^{+}{u}^{\mathrm{\prime }}$ and $v{\equiv }^{+}{v}^{\mathrm{\prime }}$, then $uv{\equiv }^{+}{u}^{\mathrm{\prime }}v{\equiv }^{+}{u}^{\mathrm{\prime }}{v}^{\mathrm{\prime }}$ by compatibility and transitivity. Associativity and the identity law descend from concatenation, so $A^{+}$ is a monoid with identity $[\epsilon ]$ (Semigroup and monoid); $A^{+}$ is the Artin monoid of $(S,m)$, and ${\sigma }_s:=[s]$ for $s\in S$.

(4) The Artin group. Let $F(S)$ be the free group on $S$ (Free group on a set of generators) and, for $s\mathrm{\ne }t$ with $m(s,t)<\mathrm{\infty }$, let ${\rho }_{s,t}:=u{v}^{-1}\in F(S)$ with $u,v$ the braid words of (2). Let $N:=⟨\text{\hspace{0.17em}⁣}⟨\{{\rho }_{s,t}:s\mathrm{\ne }t,m(s,t)<\mathrm{\infty }\}⟩\text{\hspace{0.17em}⁣}{⟩}_{F(S)}$ be their normal closure (The normal closure of a subset of a group, Relators and relations; finitely generated, finitely related, and finite presentations), and put

$A:=A(S,m):=F(S)\mathrm{/}N\mathrm{.}$

Equivalently, $A$ is the group presented by $⟨S\mid u=v\text{ for every braid pair }\{u,v\}⟩$ in the sense of Group presentation by generators and relations; its elements are the left cosets $gN$, and the images ${\sigma }_s$ of $s\in S$ generate $A$ (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$, The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups, Group and abelian group). No relation ${\sigma }_s^2=1$ is imposed: the element ${\sigma }_s^2$ need not be trivial in $A$.

(5) The canonical comparison $A^{+}\to A$. The two braid words of a braid pair have equal images in $A$, because their difference lies in $N$. Hence the assignment $s\mapsto {\sigma }_s$ is constant on braid pairs, and by minimality of ${\equiv }^{+}$ it induces a monoid homomorphism

$\gamma :A^{+}⟶A,\gamma ({\sigma }_s)={\sigma }_s,$

regarding the group $A$ as a monoid under its multiplication. It is the unique monoid homomorphism with $\gamma ({\sigma }_s)={\sigma }_s$ for all $s$. This is the construction whose universal property is proved in Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares ↗ (1); that lemma is the recorded justifier of this definition.

(6) Conventions and abstentions. (i) A value $m(s,t)=\mathrm{\infty }$ imposes no braid pair and hence no relation in either $A^{+}$ or $A$. (ii) The constructions depend only on the pair $(S,m)$. (iii) Nothing is asserted here about injectivity of $\gamma$ or of $A^{+}\to W$, about Ore localisation or a group of fractions, about torsion-freeness of $A$, about the word problem, about $A^{+}$ embedding in $A$, or about the $K(\pi ,1)$ property; in particular no multiplicativity of any lift of a reduced expression is claimed by this definition. (iv) No topological model (configuration space, hyperplane complement, Salvetti complex) is constructed or used. Part (3) establishes the congruence and quotient multiplication; Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares ↗ establishes their universal property.