Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares

For (1), define $f(s_1\cdots s_k):=f(s_1)\cdots f(s_k)$ and $f(\epsilon ):=e_M$ on $S^{\ast }$, and let $\sim$ be the relation $u\sim v:\iff f(u)=f(v)$. Then $\sim$ is an equivalence relation, and it is compatible with concatenation: if $f(u)=f(v)$ then $f(xuy)=f(x)f(u)f(y)=f(x)f(v)f(y)=f(xvy)$ for all $x,y\in S^{\ast }$. By hypothesis $\sim$ contains every braid pair, so minimality of ${\equiv }^{+}$ gives ${\equiv }^{+}\subseteq \sim$; hence $\bar{f}([u]):=f(u)$ is well defined, and $\bar{f}([u][v])=\bar{f}([uv])=f(u)f(v)=\bar{f}([u])\bar{f}([v])$ together with $\bar{f}([\epsilon ])=e_M$ makes $\bar{f}$ a monoid homomorphism with $\bar{f}({\sigma }_s)=f(s)$. Conversely every element of $A^{+}$ is a class of a word, hence a product of the elements ${\sigma }_s=[s]$, so a monoid homomorphism out of $A^{+}$ is determined by its values on the ${\sigma }_s$ and $\bar{f}$ is the unique homomorphism with $\bar{f}({\sigma }_s)=f(s)$.

The two words of a braid pair have equal images in $W$. Let $m=m(s,t)<\mathrm{\infty }$, $u=sts\cdots$ and $v=tst\cdots$ be the alternating words of length $m$. In $W$ one has $s^2=t^2=1$ and $(st{)}^m=1$ by [F14]. If $m=2k$ is even, then $u=(st{)}^k$ and $v=(ts{)}^k$; from $(st)(ts)=s{t}^{2}s=s^2=1$ one gets $ts=(st{)}^{-1}$, hence $(ts{)}^k=(st{)}^{-k}=(st{)}^{2k-k}=(st{)}^k=u$ because $(st{)}^{2k}=(st{)}^m=1$. If $m=2k+1$ is odd, then $u=(st{)}^{k}s$ and $v=(ts{)}^{k}t=(st{)}^{-k}t=(st{)}^{m-k}t=(st{)}^{k+1}t=(st{)}^{k}s=u$, using $(st)t=s$ and again $ts=(st{)}^{-1}$.

For (2), regard $S$ as a subset of the free group $F(S)$. By [F4] the map $f:S\to G$ extends uniquely to a homomorphism $\widehat{f}:F(S)\to G$ with $\widehat{f}(s)=f(s)$; on words this gives $\widehat{f}(u)=f(u)$ for the two words of any braid pair. For every braid pair, with ${\rho }_{s,t}=u{v}^{-1}$ as in [F3], one has $\widehat{f}({\rho }_{s,t})=\widehat{f}(u)\widehat{f}(v{)}^{-1}=f(u)f(v{)}^{-1}=1$, so $\widehat{f}$ kills the set $\{{\rho }_{s,t}\}$; its kernel is a normal subgroup by [F6] and hence contains the normal closure $N=⟨\text{\hspace{0.17em}⁣}⟨\{{\rho }_{s,t}\}⟩\text{\hspace{0.17em}⁣}{⟩}_{F(S)}$ by [F5]. Thus [F7] gives a homomorphism $\bar{f}:A\to G$ with $\bar{f}(gN)=\widehat{f}(g)$, and in particular $\bar{f}({\sigma }_s)=f(s)$; it is unique with this property because the elements ${\sigma }_s$ are the images of $S$ and generate $A$ in the sense of [F8]. Writing the relators as the equations $u=v$ exhibits $(A,({\sigma }_s))$ as the presented group $⟨S\mid u=v(\text{braid pairs})⟩$: a homomorphism out of this presentation is exactly a map $S\to G$ whose two values on each braid pair agree, and it is unique, which is the universal property just proved.

For (3), step 1.2 says that the map $s\mapsto s$ sends every braid pair to an equality in $W$, so (2) of step 1.3 yields $\pi :A\to W$ with $\pi ({\sigma }_s)=s$, and (1) with $M=W$ applied to the same map yields ${\pi }^{+}:A^{+}\to W$ with ${\pi }^{+}({\sigma }_s)=s$. The homomorphism $\pi$ is surjective: every element of $W$ is a product of the images of elements of $S$, and these are the images under $\pi$ of the generators ${\sigma }_s$ of $A$, so $\pi$ satisfies the surjectivity criterion of [F10]; equivalently the images $\pi ({\sigma }_s)=s$ generate $W$ by [F8]. Finally $\pi \circ \gamma ={\pi }^{+}$, because both sides are monoid homomorphisms $A^{+}\to W$ agreeing on the ${\sigma }_s$, and these generate $A^{+}$ by the uniqueness clause of step 1.1.

For (5), let $n\ge 2$ and let $m$ be the type-$A_{n-1}$ matrix on $S=\{s_1,\dots ,s_{n-1}\}$, so $m(s_i,s_j)=3$ for $\mathrm{\mid }i-j\mathrm{\mid }=1$ and $m(s_i,s_j)=2$ for $\mathrm{\mid }i-j\mathrm{\mid }>1$. Its braid pairs are then exactly the pairs of alternating words of length $3$, $\{s_i{s}_j{s}_i,s_j{s}_i{s}_j\}$ for $\mathrm{\mid }i-j\mathrm{\mid }=1$, and the commuting pairs $\{s_i{s}_j,s_j{s}_i\}$ for $\mathrm{\mid }i-j\mathrm{\mid }>1$; these are precisely the two relation families of [F11], so the identity correspondence $s_i\leftrightarrow {\sigma }_i$ matches the defining relations of $A(S,m)$ with those of $B_n^{\mathrm{A}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}}$. By (2) of step 1.3 there is a homomorphism $A(S,m)\to B_n^{\mathrm{A}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}}$ with ${\sigma }_{s_i}\mapsto {\sigma }_i$, and by [F10] applied to the presentation of [F11] there is a homomorphism $B_n^{\mathrm{A}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}}\to A(S,m)$ with ${\sigma }_i\mapsto {\sigma }_{s_i}$; the two are inverse on the generators, hence mutually inverse and so isomorphisms by [F9].

Under AC, [F12] identifies $B_n^{\mathrm{A}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{n}}$ with $B_n^{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$ by the published isomorphism ${\phi }_n$ carrying ${\sigma }_i$ to the standard geometric half twist. Composing with the isomorphism of step 2.2 identifies $A(S,m)$ with $B_n^{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$ under the stated generator correspondence. This is the only place where choice is used: the applied completeness theorem is an AC-conditional statement [F13], while the constructions and arguments of (1)--(4) and step 2.2 are explicit on finite relator sets and use no choice.

For (4), let $D_0:=\{{\sigma }_s^2:s\in S\}$, so $D=⟨\text{\hspace{0.17em}⁣}⟨D_0⟩\text{\hspace{0.17em}⁣}{⟩}_A$. Since $\pi ({\sigma }_s^2)=\pi ({\sigma }_s{)}^2=s^2=1$, each ${\sigma }_s^2$ lies in $\ker \pi$, which is normal by [F6]; hence $D\subseteq \ker \pi$ by [F5], and [F7] gives the induced homomorphism $\bar{\pi }:A\mathrm{/}D\to W$ with $\bar{\pi }({\sigma }_{s}D)=s$. For the inverse direction, put $f(s):={\sigma }_{s}D$; then $f(s{)}^2={\sigma }_s^{2}D=D$, and for $s\mathrm{\ne }t$ with $m=m(s,t)<\mathrm{\infty }$ the defining braid relation of $A$ and the relations ${\sigma }_s^{2}D={\sigma }_t^{2}D=D$ give $(f(s)f(t){)}^m=D$: writing $k:=\lfloor m\mathrm{/}2\rfloor$, the braid relation reads $({\sigma }_s{\sigma }_t{)}^k=({\sigma }_t{\sigma }_s{)}^k$ when $m=2k$, so that $({\sigma }_s{\sigma }_t{)}^{2k}=({\sigma }_s{\sigma }_t{)}^k({\sigma }_s{\sigma }_t{)}^k=({\sigma }_s{\sigma }_t{)}^k({\sigma }_t{\sigma }_s{)}^k=({\sigma }_s{\sigma }_t{)}^{k-1}{\sigma }_s{\sigma }_t{\sigma }_t{\sigma }_s({\sigma }_t{\sigma }_s{)}^{k-1}=\cdots =1$ in $A\mathrm{/}D$ by ${\sigma }_s^2={\sigma }_t^2=1$, while for $m=2k+1$ it reads $({\sigma }_s{\sigma }_t{)}^k{\sigma }_s=({\sigma }_t{\sigma }_s{)}^k{\sigma }_t$, so that $({\sigma }_s{\sigma }_t{)}^{2k+1}=({\sigma }_s{\sigma }_t{)}^k{\sigma }_s{\sigma }_t({\sigma }_s{\sigma }_t{)}^k=({\sigma }_t{\sigma }_s{)}^k{\sigma }_t^2({\sigma }_s{\sigma }_t{)}^k=({\sigma }_t{\sigma }_s{)}^k({\sigma }_s{\sigma }_t{)}^k=D$. The universal property of $W$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups) therefore gives a homomorphism $\phi :W\to A\mathrm{/}D$ with $\phi (s)={\sigma }_{s}D$.

The homomorphisms $\bar{\pi }$ and $\phi$ of step 3.2 are mutually inverse: $\bar{\pi }(\phi (s))=\bar{\pi }({\sigma }_{s}D)=s$ for all $s$, and $\phi (\bar{\pi }({\sigma }_{s}D))=\phi (s)={\sigma }_{s}D$ for all $s$; two homomorphisms out of a group generated by a set are equal when they agree on that set, while $W$ is generated by the images of $S$ and $A\mathrm{/}D$ is generated by the ${\sigma }_{s}D$ by [F8]. Hence $\bar{\pi }$ is bijective, i.e. an isomorphism by [F9]. Since $\pi$ is the composite of the quotient map $A\to A\mathrm{/}D$ with $\bar{\pi }$ and $\bar{\pi }$ is injective, an element $g\in A$ lies in $\ker \pi$ exactly when $gD=D$, that is, exactly when $g\in D$; thus $\ker \pi =D$, and imposing the relations ${\sigma }_s^2=1$ on the Artin presentation returns the Coxeter presentation in the sense of [F7] and [F10].

Scope: nothing in the proof establishes injectivity of $\gamma$ or of ${\pi }^{+}$, torsion-freeness of $A$, solvability of the word problem, an Ore localisation, an embedding of $A^{+}$ into $A$, or a general $K(\pi ,1)$ statement, and the only identification with geometric braids is the conditional application of step 3.1.