On representations of Artin groups and the Tits conjecture
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.
subgroups of automorphisms of polynomial algebras
scientific article; zbMATH DE number 701280
scientific article