Symmetries and pseudo-symmetries of hyperelliptic surfaces

Abstract

1. Let X be a closed Riemann surface of genus g ≥ 2 and let Aut X denote the group of automorphisms of X where, in this paper, an automorphism means a conformal or anticonformal self-homeomorphism. X is called hyperelliptic if it admits a conformal automorphism J of order 2 such that X/H has genus 0, where H = 〈J〉 is the group of order 2 generated by J. Thus X is a two-sheeted covering of the sphere which is branched over 2g + 2 points and J is the sheet-interchange map. J is the unique conformal automorphism of order 2 such that X/〈J〉 has genus 0 and it follows that if U ∈ Aut X, then UJU−1 = J. Thus J is central in Aut X and H ≤ Aut X. (Cf. [8])