ρ-hyperelliptic-symmetric schottly groups*

Abstract

Let S be a γ-hyperelliptic Riemann surface, with a γ-hyperelliptic involution τ. Assume that S has a symmetry σ so that στ=τσ. If H denotes the group generated by τ and σ, then we show that H is of Schottky type, that is, there is a Schottky uniformization (ωG P: ω S) of S for which the group H lifts. For hyperelliptic Riemann surfaces, we describe explicitly Schottky uniformizations (hyperelliptic-symmetric Schottky groups) for which both τ and σ lift. The particularity of these uniformizations is that both τ and σ are reflected in a marking of the uniformizing groups. For g=2 we use the above groups to describe inside the Schottky space of genus two, the locus of symmetric Riemann surfaces.