Mühlherr's twist conjecture for finite-rank Coxeter systems

Let (W,S)(W,S) be a Coxeter system of finite rank, and let SS' be another set of Coxeter generators of WW. Define

SW={wsw1:sS and wW}.S^W=\{wsw^{-1}:s\in S\text{ and }w\in W\}.

The set SS is sharp-angled with respect to SS' if, for each pair s,tSs,t\in S such that 2<m(s,t)<2<m(s,t)<\infty, there is a wWw\in W such that w{s,t}w1Sw\{s,t\}w^{-1}\subseteq S'. The Coxeter systems (W,S)(W,S) and (W,S)(W,S') are twist equivalent if there is a finite sequence of elementary twists transforming SS into SS'.

Twist conjecture. If SSWS'\subseteq S^W and SS is sharp-angled with respect to SS', then (W,S)(W,S) is twist equivalent to (W,S)(W,S').

This conjecture concerns the isomorphism problem for finite-rank Coxeter systems and predicts that the stated inclusion and sharp-angledness conditions are sufficient for the two generating sets to be related by elementary twists. It is attributed to Mühlherr; the supplied source gives no resolution, so its status remains open.

Sources & referencesView supporting material

Primary source

John Ratcliffe and Steven Tschantz, “JSJ decompositions of Coxeter groups over FA subgroups”, arXiv:0910.5732 (2009).

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