Generating the genus g+1 Goeritz group of a genus g handlebody

TL;DR

This paper identifies a specific generating set for the Goeritz group of a genus g+1 handlebody, aligning with Powell's conjecture, and provides two proofs—classical and thin position-based.

Contribution

It introduces a new generating set for the Goeritz group of genus g+1 handlebodies and confirms its consistency with Powell's conjecture through two distinct proofs.

Findings

The set of 4g+1 elements generates the Goeritz group.

The generators align with Powell's proposed set.

Two different proof techniques validate the results.

Abstract

A specific set of 4g+1 elements is shown to generate the Goeritz group of the genus g+1 Heegaard splitting of a genus g handlebody. These generators are consistent with Powell's proposed generating set for the Goeritz group of the genus g+1 splitting of S^3. There are two proofs: one using purely classical techniques and one using thin position.