On p-hyperelliptic Involutions of Riemann Surfaces

Abstract. A compact Riemann surface X of genus g> 1 is said to be p-hyperelliptic if X admits a conformal involution ρ, called a p-hyperelliptic invo-lution, 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