The Powell conjecture and reducing sphere complexes

Abstract

The Powell conjecture offers a finite generating set for the genus g Goeritz group, the group of automorphisms of S 3 that preserve a genus g Heegaard surface Σ g , generalizing a classical result of Goeritz in the case g = 2 . We study the relationship between the Powell conjecture and the reducing sphere complex R ( Σ g ) , the subcomplex of the curve complex C ( Σ g ) spanned by the reducing curves for the Heegaard splitting. We prove that the Powell conjecture is true if and only if R ( Σ g ) is connected. Additionally, we show that reducing curves that meet in at most six points are connected by a path in R ( Σ g ) ; however, we also demonstrate that even among reducing curves meeting in four points, the distance in R ( Σ g ) between such curves can be arbitrarily large. We conclude with a discussion of the geometry of R ( Σ g ) .