A proof that all Seifert 3-manifold groups and all virtual surface groups are conjugacy separable (Q2370278)

A subset \(S\) of a group \(G\) is called \textit{separable} if, for every \(g\) in \(G-S\), there is a finite quotient of \(G\) in which \(g\) and \(S\) have disjoint images. A group \(G\) is called \textit{conjugacy separable} if conjugacy classes are separable. In \textit{R. B. J. T. Allenby, G. Kim}, and \textit{C. Y. Tang} [J. Algebra 285, 481--507 (2005; Zbl 1077.57001)], certain Seifert \(3\)-manifold groups are proved to be conjugacy separable: in the present paper the result is extended to \textit{all} Seifert manifold groups, by means of quite different arguments. In fact, the proof that all Seifert manifold groups are conjugacy separable given by the author is based on the algebraic structure of these groups as extensions, rather than trying to decompose them into amalgamated free products; this approach allows for an essentially unified treatment of these groups. The main tool used in the proof is the following result: if \(G\) is the fundamental group of a Seifert fibred space, then there exists a short exact sequence \(1\to C \to G \to H \to 1\), where \(C\) is cyclic and \(H\) is a virtually surface group (i.e. \(H\) contains a finite index subgroup isomorphic to a surface group).