SCHOTTKY UNIFORMIZATIONS AND RIEMANN MATRICES OF MAXIMAL SYMMETRIC RIEMANN SURFACES OF GENUS 5∗

In this note we consider pairs (S, τ), where S is a closed Riemann surface of genus five and τ: S → S is some anti-conformal involution with fixed points so that K(S, τ) = {h ∈ Aut±(S) : hτ = τh} has the maximal order 96 and S/τ is ori-entable. We observe that there are exactly two topologically dif-ferent 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 Rie-mann matrices