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We show that if the monodromy of an open book decomposition has sufficientlyhigh displacement distance, acting on the loop and arc complex for a page, thenit is the unique minimal Euler characteristic open book for the manifold. Inparticular, we show that such an open book induces the unique (up to isotopy)minimal genus Heegaard surface for the manifold, and that this Heegaard surfacehas cyclic mapping class group.

Heegaard splittings and open books