A small generating set for the balanced superelliptic handlebody group

TL;DR

This paper establishes minimal generating sets for the balanced superelliptic handlebody group and related groups, advancing understanding of their algebraic structure with explicit generators.

Contribution

It provides the first known minimal generating sets for the balanced superelliptic handlebody group and the liftable Hilden group, including proofs of minimality in several cases.

Findings

The balanced superelliptic mapping class group is generated by four elements.

The liftable Hilden group is generated by three elements.

These generating sets are minimal in most cases.

Abstract

The balanced superelliptic handlebody group is the normalizer of the transformation group of the balanced superelliptic covering space in the handlebody group of the total space. We prove that the balanced superelliptic mapping class group is generated by four elements. To prove this, we also proved that the liftable Hilden group is generated by three elements. This generating set for the liftable Hilden group is minimal except for some hyperelliptic cases and the generating set for the balanced superelliptic mapping class group above is also minimal for several cases.