Artin monoids inject in their groups.

*(English)*Zbl 1020.20026Let \(M=(m_{ij})\) 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 \(\mathbb{Q}(x,y)\).

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 \(\mathbb{Q}(x,y)\).

Reviewer: Carl Droms (Harrisonburg)