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Artin monoids inject in their groups. (English) Zbl 1020.20026
Let $$M=(m_{ij})$$ be an $$n\times n$$ symmetric matrix such that $$m_{ii}=1$$ for each $$i$$ and $$m_{ij}\in\{2,3,4,\dots,\infty\}$$ whenever $$i\neq j$$. $$M$$ defines a so-called ‘Coxeter graph’ $$\Gamma$$ having $$n$$ vertices $$v_1,\dots,v_n$$, with (i) no edge between $$v_i$$ and $$v_j$$ if $$m_{ij}=2$$, (ii) an unlabeled edge between them if $$m_{ij}=3$$, and (iii) an edge labeled by $$m_{ij}$$ in all other cases.
To $$\Gamma$$ we associate the presentation $$P=\langle a_1,\dots,a_n\mid\text{prod}(a_i,a_j,m_{ij})=\text{prod}(a_j,a_i,m_{ij})\rangle$$ (where $$\text{prod}(x,y,m)$$ denotes the product $$xyx\cdots$$ having a total of $$m$$ factors). The group $$G_\Gamma$$ with presentation $$P$$ is called an ‘Artin group’, and the monoid $$G_\Gamma^+$$ with presentation $$P$$ is called an ‘Artin monoid’.
It was previously known that the natural (monoid) homomorphism $$G_\Gamma^+\to G_\Gamma$$ was one-to-one for certain Artin groups (those of finite-type, for example), but it was unknown if this was true in general. In the present paper, the author proves that it is one-to-one in all cases.
By the definitions of $$G_\Gamma$$ and $$G_\Gamma^+$$, it suffices to prove that $$G_\Gamma^+$$ is a submonoid of any group. He does this by first proving that for any Coxeter graph $$\Gamma$$, there is another one, $$\widetilde\Gamma$$, of small type (i.e., all $$m_{ij}\in\{2,3\}$$) and triangle-free, such that $$G_\Gamma^+$$ injects into $$G_{\widetilde\Gamma}^+$$. He then proves that for any triangle-free Coxeter graph $$\Gamma$$ of small type, there is a faithful representation $$G_\Gamma^+\to\text{GL}(V)$$ where $$V$$ is a (possibly infinite-dimensional) vector space over $$\mathbb{Q}(x,y)$$.

##### MSC:
 20F36 Braid groups; Artin groups 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20M05 Free semigroups, generators and relations, word problems
##### Keywords:
Artin groups; Artin monoids; faithful representations
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