Schottky Uniformizations of Genus 6 Riemann Surfaces Admitting A5 as Group of Automorphisms

In this note we construct a 1-complex dimensional family of (marked) Schottky groups of genus 6 with the property that every closed Riemann surface of genus 6 admitting the group A₅ as conformal group of automorphisms is uniformized by one of these Schottky groups. In the algebraic limit closure of this family we describe three noded Schottky groups uniformizing the three boundary points of the pencil described by González-Aguilera and Rodriguez. We are able to find a very particular Riemann surface of genus 6 which is a (local) extremal for a maximal set of homologically independent simple closed geodesics. We observe that it is not Wimann's curve, the only Riemann surface of genus 6 with S₅ as group of conformal automorphisms. The Schottky uniformizations permit us to compute a reducible symplectic representation of A₅.