Abstract. We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

Let M be a compact oriented irreducible 3-manifold which is neither a graph manifold nor a hyperbolic manifold. We prove that π1M is 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 with boundary, thus proving that their fundamental groups are virtually special, and hence linear.