On p-hyperelliptic Involutions of Riemann Surfaces

A compact Riemann surface X of genus g > 1 is said to be phyperelliptic if X admits a conformal involution ρ, called a p-hyperelliptic involution, for which X/ρ is an orbifold of genus p. Here we give a new proof of the well known fact that for g > 4p + 1, ρ is unique and central in the group of all automorphisms of X. Moreover we prove that every two p-hyperelliptic involutions commute for 3p + 2 ≤ g ≤ 4p + 1 and X admits at most two such involutions if g > 3p+2. We also find some bounds for the number of commuting p-hyperelliptic involutions and general bound for the number of central p-hyperelliptic involutions.