On rigidity of Coxeter systems up to finite twists and separations of Coxeter generating sets

Abstract: In this paper, we study the twist-conjecture for Coxeter systems and rigidity of Coxeter systems up to finite twists. For Coxeter systems $(W,R)$ and $(W,S)$, under the untangle-condition for conjugate subsets, we investigate separations and type(I) and type(II) subsets of $R$ and $S$ and give an equivalent condition of $R$ and $S$ that are conjugate up to finite twists. We provide one direction of approach to solving the twist-conjecture and the isomorphism problem for Coxeter groups of finite ranks.