On representations of Artin groups and the Tits conjecture (Q1340254)

Let \(M = (m_{ij})\) denote an \(n \times n\) Coxeter matrix \((m_{ij} = m_{ji} \in \mathbb{Z}^{\geq 2} \cup \{\infty\}\) for \(i \neq j\) and \(m_{ii} = 1\)). Let \(A(M)\) be an Artin group with generating set \(\{a_ 1, \dots, a_ n\}\) and for each \(i \neq j\) a relation \(a_ i a_ j a_ i \dots = a_ j a_ i a_ j \dots\), where both sides are words of length \(m_{ij}\). If \(m_{ij} = 2\) then \(a^ 2_ i\) and \(a^ 2_ j\) commute. The Tits conjecture [stated by \textit{K. I. Appel} and \textit{P. E. Schupp}, Invent. Math. 72, 201-220 (1983; Zbl 0536.20019)] says that for an arbitrary Artin group the only relations between the \(a^ 2_ i\) are these commutator relations. \(A(M)\) is said to be of small type if \(m_{ij} \leq 3\) for all \(i\), \(j\). The author proves that if \(A(M)\) is of small type, then (similar to a braid group) \(A(M)\) can be represented as a subgroup of \(\text{Aut} (F_ n)\) and as a subgroup of automorphisms of a polynomial algebra. By the use of these representations the author proves the Tits conjecture for braid groups and certain other Artin groups of small type.