A family of $(p,n)$-gonal Riemann surfaces with several $(p,n)$-gonal groups

TL;DR

This paper constructs a family of high-genus Riemann surfaces with multiple $(p,n)$-gonal groups, expanding understanding of their symmetry properties for prime $p$ and integer $n$.

Contribution

It introduces a new family of $(p,n)$-gonal Riemann surfaces with maximal genus that possess multiple such groups, highlighting their complex symmetry structures.

Findings

Constructed a family of maximal genus $(p,n)$-gonal Riemann surfaces

Demonstrated the existence of multiple $(p,n)$-gonal groups on these surfaces

Extended the classification of symmetries in Riemann surfaces

Abstract

Let $p \geqslant 3$ be a prime number and let $n \geqslant 0$ be an integer such that $p-1$ divides $n.$ In this short note we construct a family of $(p,n)$-gonal Riemann surfaces of maximal genus $2np+(p-1)^2$ with more than one $(p,n)$-gonal group.