Graph manifolds with boundary are virtually special (Q2928217)

This paper considers separability properties of surfaces properly embedded in graph manifolds. Recall that a subgroup \(F\) of a group \(G\) is separable if, for each \(g\in G-F\), there is a finite index subgroup \(H\) of \(G\) with \(g\notin HF\). If \(F_1\), \(F_2\) are subgroups of \(G\), the double coset \(F_1F_2\) is separable if, for each \(g\in G-F_1F_2\), there is a finite index subgroup \(H\) of \(G\) with \(g\notin HF_1F_2\). A graph manifold is an oriented compact connected irreducible 3-manifold that has only Seifert-fibred pieces in its \(JSJ\) decomposition. The main results of the paper are the following: a) Let \(M\) be a graph manifold, with or without boundary, and let \(S\) be an oriented incompressible surface properly embedded in \(M\). Then \(\pi_1(S)\) is separable in \(\pi_1(M)\). b) Let \(S, P\subset M\), be oriented incompressible surfaces whose intersection with each block of the \(JSJ\) decomposition of \(M\) is horizontal or vertical. Let \(\tilde S,\tilde P\) be intersecting components of the preimages of \(S,P\) in the universal cover \(\tilde M\) of \(M\). Then \(Stab(\tilde S)Stab(\tilde P)\) is separable in \(\pi_1(M)\).NEWLINENEWLINELet \(M\) be a graph manifold containing a sufficient collection of incompressible oriented surfaces, that is, a collection so that the intersection of each surface with each block of the \(JSJ\) decomposition of \(M\) is vertical or horizontal, and so that for each block \(B\) and each torus in \(\partial B\), there is a surface intersecting them, which is vertical with respect to \(B\). Using the main results it is proved that \(\pi_1(M)\) is virtually the fundamental group of a special cube complex. Then it is shown that a graph manifold with boundary always has a sufficient collection, implying then that if \(M\) is a graph manifold with non-empty boundary, then \(\pi_1(M)\) is virtually special. See [\textit{F. Haglund} and \textit{D. T. Wise}, Geom. Funct. Anal. 17, No. 5, 1551--1620 (2008; Zbl 1155.53025)] for definitions and facts about special cube complexes.