The Heegaard genus of amalgamated 3-manifolds

Let M and M’ be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that M and M’ are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is sufficiently complicated', the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of M and M' and the genus of their common boundary. Here, a homeomorphism is sufficiently complicated’ if it is the composition of a homeomorphism from the boundary of M to some surface S, followed by a sufficiently high power of a pseudo-Anosov on S, followed by a homeomorphism to the boundary of M’. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.