Finite presentations for the balanced superelliptic mapping class groups

TL;DR

This paper provides finite presentations for the balanced superelliptic mapping class groups of various surfaces, enhancing understanding of their algebraic structure and relations.

Contribution

It introduces new finite presentations for these groups, differing from previous work, and applies them to surfaces with different topological features.

Findings

Finite presentations for closed surfaces.

Finite presentations for surfaces with one marked point.

Finite presentations for surfaces with one boundary component.

Abstract

The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering space in the mapping class group of the total surface. We give finite presentations for the balanced superelliptic mapping class groups of closed surfaces, surfaces with one marked point, and surfaces with one boundary component. To give these presentations, we construct finite presentations for corresponding liftable mapping class groups in a different generating set from Ghaswala-Winarski's presentation in \cite{Ghaswala-Winarski1}.