Powell moves and the Goeritz group

Abstract

In 1980 J. Powell proposed that five specific elements sufficed to generate the Goeritz group of any Heegaard splitting of $S^3$, extending work of Goeritz on genus $2$ splittings. Here we prove that Powell's conjecture was correct for splittings of genus $3$ as well, and discuss a framework for deciding the truth of the conjecture for higher genus splittings.

39 pages; 17 figures

References in corpus (1)

• Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting

Cited by in corpus (4)

• Describing elements of the genus-2 Goeritz group of $S^3$
• Goeritz groups of bridge decompositions
• Thick isotopy property and the mapping class groups of Heegaard splittings
• Twisted book decompositions and the Goeritz groups