Topological classification of conformal actions on elliptic-hyperelliptic Riemann surfaces

A compact Riemann surface X of genus g>1 is said to be elliptic-hyperelliptic if X admits a conformal involution ρ for which X/ρ is an orbifold of genus 1. Here we classify conformal actions on elliptic-hyperelliptic Riemann surfaces of genus g>5 up to topological conjugacy and determine which of them can be maximal.