A finiteness property and an automatic structure for Coxeter groups
If \(\alpha\) and \(\beta\) are positive roots in the root system of a Coxeter group \(W\), we say that \(\alpha\) dominates \(\beta\) if \(w\cdot\beta\) is negative for every \(w\in W\) such that \(w\cdot\alpha\) is negative. It is shown that if the distinguished generating set of \(W\) is finite, then the set of all positive roots \(\alpha\) for which there is no \(\beta\) dominated by \(\alpha\) is finite. It follows as a consequence that the group \(W\) is automatic.
scientific article; zbMATH DE number 537293