Virtually unipotent curves in some non-NPC graph manifolds

Graph manifolds with boundary are virtually special

Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in π1M , and that the double cosets for crossing surfaces are also separable. We deduce that if there is a ‘sufficient’ collection of surfaces in M , then π1M is virtually the fundamental group of a special non-positively curved cube complex. We provide a sufficient collection for ...

Graph manifolds have virtually positive Seifert volume

This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold N , the set of mapping degrees 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.

Immersed and virtually embedded π1-injective surfaces in graph manifolds

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed 1 -injective surface of negative Euler characteristic. We also determine the class of closed graph manifolds which have no nite cover containing an embe...

Mixed 3–manifolds Are Virtually Special

LetM be a compact oriented irreducible 3–manifold which is neither a graph manifold nor a hyperbolic manifold. We prove that π1M is virtually special.

Graph decomposition is NPC - A complete proof