Graph manifolds have virtually positive Seifert volume (Q2909040)

The present paper deals with the invariant Seifert volume for 3-manifolds, which vanishes (resp. is positive) for closed 3-manifolds supporting the geometries \(\mathbb S^3\), \(\mathbb S^2 \times \mathbb R\), \(\mathbb R^3\), Nil, \(\mathbb H^2 \times \mathbb R\) and Sol (resp. the geometry \( \widetilde {SL_2}(\mathbb R)\)) and seems to reflect the gluing information of the JSJ-pieces for 3-manifolds with non-trivial JSJ-decomposition: see [\textit{R. Brooks} and \textit{W. Goldman}, Trans. Am. Math. Soc. 286, 651--664 (1984; Zbl 0548.57016) and Duke Math. J. 51, 529--545 (1984; Zbl 0546.57003)].NEWLINENEWLINEA graph manifold (i.e. a prime 3-manifold whose JSJ-pieces are all Seifert manifolds) is called trivial if it is covered by either a torus bundle or a Seifert manifold. The main theorem of the present paper states that, \textit{for any closed non-trivial graph manifold \(N\), there exists a finite covering \(\tilde N\) of \(N\) whose Seifert volume is positive}.NEWLINENEWLINEThe authors apply the theorem to the study of mapping degrees.NEWLINENEWLINEAs a consequence they obtain a complete answer for prime 3-manifolds to a well-known problem (see Problem A of \textit{A. Reznikov} [Math. Ann. 306, No. 3, 547--554 (1996; Zbl 0859.20027)] or Question 1.3 of [\textit{S. C. Wang}, in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 457--468 (2002; Zbl 1009.57025)]): for each closed orientable prime 3-manifold \(N\), the set of mapping degrees \(\mathcal D(M,N)\) is finite for any 3-manifold \(M\), unless \(N\) is finitely covered by either a torus bundle, or a trivial circle bundle, or the 3-sphere.