Separability criterion for graph-manifold groups (Q1292726)

Let \(S\to M\) be an immersion of a compact connected surface \(S\) in an irreducible compact \(3\)-manifold \(M\). Recall that the immersion is said to be incompressible if the corresponding morphism of fundamental groups is injective. If moreover \(S\) lifts up to homotopy to an embedded surface in some finite cover of \(M\), then \(S\) is said to be separable. This is e.g. always the case if the fundamental group of \(M\) is subgroup separable. The paper under review states and investigates the following conjecture based on a result of \textit{J. H. Rubinstein} and \textit{Sh. Wang} [Comment. Math. Helv. 73, No. 4, 499-515 (1998; Zbl 0916.57001)]: Conjecture: Suppose \(S\to M\) is incompressible and \(M\) is a graph manifold obtained by gluing Seifert fibre spaces along boundary tori. Then \(S\) is separable if and only if certain fibre intersection ratios are equal to one. For horizontal immersions \(S\to M\), this statement has been proved by Rubinstein and Wang [loc. cit.]. The paper under review extends this result to non-horizontal surfaces under certain restrictions on \(M\). First of all, \(M\) is supposed to be good, which means that the base orbifolds of the Seifert fibre spaces have a smooth universal cover. Now the conjecture is proved in the following three settings: (1) The gluing graph is a tree. (2) There are only two Seifert fibre spaces in presence. (3) There is only one Seifert space in presence and its boundary components are glued pairwise.