A finiteness result for Heegaard splittings

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of 3g−3 disjoint essential disks. These are characterized by a combinatorial criterion which is a slight strengthening of Casson–Gordonʹs rectangle condition. This is in contrast to fact that in general there can exist infinitely many such systems of disks which satisfy just the Casson–Gordon rectangle condition.