Heegaard structure respects complicated JSJ decompositions

Let M be a 3-manifold with torus boundary components T 1 and T 2 . Let ϕ : T 1 → T 2 be a homeomorphism, M ϕ the manifold obtained from M by gluing T 1 to T 2 via the map ϕ , and T the image of T 1 in M ϕ . We show that if ϕ is “sufficiently complicated” then any incompressible or strongly irreducible surface in M ϕ can be isotoped to be disjoint from T . It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.