The balanced superelliptic mapping class groups are generated by three elements

TL;DR

This paper proves that balanced superelliptic mapping class groups, with various boundary and marked point configurations, are generated by three elements, and these generating sets are minimal in most cases.

Contribution

It establishes that these groups are generated by three elements and shows the minimality of these generating sets in most scenarios.

Findings

Balanced superelliptic mapping class groups are generated by three elements.

Liftable mapping class groups are also generated by three elements.

The minimality of these generating sets is confirmed in most cases.

Abstract

The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering in the mapping class group of the total surface. We prove that the balanced superelliptic mapping class groups with either one marked point, one boundary component, or no marked points and boundary are generated by three elements. To prove this, we also show that its liftable mapping class groups are also generated by three elements. These generating sets are minimal except for several no marked points and boundary cases.