Abstract
In this note we consider pairs (S, τ ), where S is a closed Riemann surface of genus five and τ : S → S is some anticonformal involution with fixed points so that K(S, τ ) = {h ∈ Aut±(S) : hτ = τh} has the maximal order 96 and S/τ is orientable. We observe that there are exactly two topologically different choices for τ . They give non-isomorphic groups K(S, τ ), each one acting topologically rigid on the respective surface S. These two cases give then two (connect) real algebraic sets of real dimension one in the moduli space of genus 5. In this note we describe these components by classical Schottky groups and with the help of these uniformizations we compute their Riemann matrices.
Highlights
A closed Riemann surfaces S of genus g ≥ 2 is called symmetric if it admits an anticonformal involution τ : S → S, called a symmetry of S
We say that (S, τ ) is maximal symmetric if the maximal order 24(g − 1) is attained for K(S, τ ); the surface S is said to be maximal symmetric and the reflection τ to be a maximal reflection
It is a well known fact that any closed symmetric Riemann surface S with a reflection τ : S → S can be uniformized by a classical Schottky group [7], [16], [21] and [24]
Summary
A closed Riemann surfaces S of genus g ≥ 2 is called symmetric if it admits an anticonformal involution τ : S → S, called a symmetry of S. For a maximal symmetric Riemann surface S with a maximal reflection τ : S → S we must have that S/K(S, τ ) is a closed disc with exactly four branched values of orders 2, 2, 2 and 3 on its border. The quotient S/H is a closed Riemann surface of genus two admitting the group K(S, τ )/H as group of automorphisms This group is isomorphic to Z2+Z2+D3, where D3 is a conformal group isomorphic to the dihedral group of order 6, one of the Z2 is generated by the hyperelliptic involution and the other Z2 is generated by the reflection induced by τ. We describe some algebraic limits giving a couple of noded Schottky groups These noded Schottky groups uniformize boundary points in Moduli space of the above Riemann surfaces with automorphism. Burnside [2] to compute the Riemann matrices of these surfaces
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