On conjugacy separability of some Coxeter groups and parabolic-preserving automorphisms. (Q405403)

A group \(G\) is called conjugacy separable if for any two non-conjugate elements \(x,y\in G\) there is a homomorphism from \(G\) to a finite group \(M\) such that the images of \(x\) and \(y\) are not conjugate in \(M\); or equivalently if each conjugacy class \(x^G:=\{gxg^{-1}\mid g\in G\}\) is closed in the profinite topology on \(G\).NEWLINENEWLINE The authors study conjugacy separability and related properties for Coxeter groups. Among the main results that the authors obtain are the following two theorems:NEWLINENEWLINE 1. If \(W\) is an even Coxeter group of finite rank such that its Coxeter diagram has no \((4,4,2)\)-triangle, then \(W\) is conjugacy separable.NEWLINENEWLINE 2. If \(W\) is a finitely generated Coxeter group with Coxeter generating set \(S\), and if \(\alpha\in\Aut(W)\) is an automorphism, then \(\alpha\) is inner-by-graph if and only if it satisfies the following two conditions: (1) \(\alpha\) maps every parabolic subgroup to a parabolic subgroup. (2) For all \(s,t\in S\), such that \(st\) has finite order in \(W\), there is a pair \(s',t'\in S\) such that \(\alpha(st)\) is conjugate to \(s't'\).