Piotr Przytycki: Graph manifolds with boundary are virtually special

This is joint work with Daniel T. Wise. Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in pi_1(M) and a similar result for pairs of crossing surfaces.

We deduce that if there is a "sufficient" collection of embedded surfaces in M, then pi_1(M) is virtually the fundamental group of a "special" CAT(0) cube complex. That is a complex that admits a local isometry into the Salvetti complex of a right-angled Artin group.

We provide a sufficient collection for graph manifolds with boundary thus proving that their fundamental groups are virtually special, in particular linear over Z.