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

TL;DR

This paper investigates the rigidity of Coxeter systems under finite twists, providing conditions for conjugacy of generating sets and advancing the twist-conjecture and isomorphism problem for finite rank Coxeter groups.

Contribution

It introduces an approach to the twist-conjecture, analyzing separations and subsets of Coxeter generating sets under the untangle-condition.

Findings

Characterization of conjugate Coxeter generating sets up to finite twists

Conditions for separations and subset types in Coxeter systems

Progress towards solving the twist-conjecture and isomorphism problem

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.