Goeritz Groups of Genus Two and Genus Three Heegaard Splitting of the Three Sphere

Abstract

In this work, we present a finite generating set (j2 of J-li, the genus-2 Goeritz group of S3, in terms of Dehn twists about certain simple closed curves on the standard Heegaard surface. We present an algorithm that describes an element f EJ-12 as a word in the alphabet of (j 2 in a certain format. Using a complexity measure defined on reducing spheres, we show that such a description off eJ-li is unique. We also present a finite subset (j3 of Jf3, the genus-3 Goeritz group of S3. We show that the elements in (h generates the generating elements of J{j proposed by Freedman and Scharlemann. Thus, we verify that (j 3 is a generating set of_J-6