Diagrammatic criteria for strong irreducibility of Heegaard splittings and finiteness of Goeritz groups

Abstract

We give two criteria for diagrams of Heegaard splittings of 3-manifolds. Weaker one of them guarantees that the splitting is strongly irreducible, and the stronger one guarantees in addition that the Goeritz group is finite. They are generalizations of the criteria introduced by Casson-Gordon and Lustig-Moriah. Their original ones require as input the diagram formed by a pair of maximal disk systems. Our present ones accept as input that of arbitrary disk systems, including minimal disk systems. In other words, our criteria accept Heegaard diagrams.