Conjugacy separability of certain Seifert 3-manifold groups

Abstract

It is well known that Seifert 3-manifold groups are residually finite. Niblo [J. Pure Appl. Algebra 78 (1992) 77-84] showed that they are double coset separable. Applying this result we show that, except for some special cases, most of the Seifert 3-manifold groups are conjugacy separable.  2004 Elsevier Inc. All rights reserved.

Conjugacy separability of Seifert 3-manifold groups over non-orientable surfaces

“…For Seifert 3-manifolds with non-trivial boundary, a positive conclusion follows immediately form work of Ribes, Segal and Zalesskii [11]. In the case of closed Seifert 3-manifolds, the fact that almost all Seifert 3-manifold groups with orientable surfaces (briefly, Seifert groups over orientable surfaces) are conjugacy separable was shown in [2]. In this paper we complete the picture by showing all Seifert groups over non-orientable surfaces are also conjugacy separable.…”

“…Niblo (1992) [9] improved this result by showing that these groups are double coset separable. In Allenby, Kim and Tang (2005) [2] it was shown that all but two types of groups in the orientable case are conjugacy separable. Martino (2007) [7] using topological results showed that Seifert groups are conjugacy separable.…”

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

“…Additionally, these properties are related to the problem of lifting immersed subspaces of topological spaces to embedded subspaces of finite covers. Recently, [1] have proved that certain Seifert 3-manifold groups are conjugacy separable and in this paper we prove that all Seifert 3-manifold groups are conjugacy separable.…”

“…where [1] φ i is understood to be a twisted-φ i conjugacy class in S. By hypothesis, each [1] φ i is closed in S and thence in G. Thus each g i [1] φ i is closed in G and thus so is the (finite) union. Thus we have shown that every conjugacy class in G is closed, which is another way of saying that G is conjugacy separable.…”

Conjugacy Separability of Generalized Free Products of Finitely Generated Nilpotent Groups

“…However generalized free products of polycyclic-by-finite groups, amalgamating central subgroups, are conjugacy separable [8]. Recently, Allenby, Kim, and Tang [1] considered the case when the amalgamated subgroup is a direct product of two cyclic groups and showed that most of Seifert groups are conjugacy separable.…”