Two aspects of graph 3-manifold groups

We prove that fundamental groups of graph 3-manifolds are virtually poly-free and lie in the family Lex. As a consequence, we prove that all finitely generated 3-manifold groups also have these two properties. The first property is a purely group-theoretical concept, and the second is related to the left-exactness property of bounded cohomology of groups. Both properties are proved by constructing sequences of covers of graph 3-manifolds.