Occurrence problem in Artin groups of finite type (Q789512)

An Artin group is a group G with generators \(a_ i\) and relations \(a_ ia_ ja_ i...=a_ ja_ ia_ j...\) for certain i and j, where both words consist of \(m_{ij}\) letters alternately \(a_ i\) and \(a_ j\). These groups include the braid groups. The associated Coxeter group \(\bar G\) is obtained from G by adding the relations \(a^ 2\!_ i=1\) for all i. If \(\bar G\) is finite, the original group G is called an Artin group of finite type. \textit{E. Brieskorn} and \textit{K. Saito} [Invent. Math. 17, 245-271 (1972; Zbl 0243.20037)] showed that the word and conjugacy problems for such groups are solvable. The present paper gives an algorithm that solves the occurrence (or membership) problem for a cyclic subgroup in an Artin group of finite type. The occurrence problem for an arbitrary finitely generated subgroup in such a group is unsolvable.