Hyperbolic geometry and the moduli space of real binary sextics, submitted

Abstract. The moduli space of real 6-tuples in CP 1 is modeled on a quotient of hyperbolic 3-space by a nonarithmetic lattice in Isom H3. This is partly an expository note; the rst part of it is an introduction to orbifolds and hyperbolic re ection groups. These notes are an exposition of the key ideas behind our result that the moduli space Ms of stable real binary sextics is the quotient of real hyperbolic 3-space H3 by a certain Coxeter group (together with its diagram automorphism). We hope they can serve as an aid in understanding our work [3] on moduli of real cubic surfaces, since exactly the same ideas are used, but the computations are easier and the results can be visualized. These notes derive from the rst author's lectures at the summer school \Algebra and Geometry around Hypergeometric Functions", held at Galatasary University in Istanbul in July 2005. He is grateful to the organizers, fellow speakers and students for making the workshop very rewarding. To keep the avor of lec-ture notes, not much has been added beyond the original content of the lectures; some additional material appears in an appendix. The pictures are hand-drawn to encourage readers to draw their own. Lecture 1 Hyperbolic space H3 is a Riemannian manifold for which one can write down an explicit metric, but for us the following model will be more useful; it is called the upper half-space model. Its underlying set is the set of points in R3 with positive vertical coordinate, and geodesics appear either as vertical half-lines, or First author partly supported by NSF grant DMS 0231585. Second and third authors partly supported by NSF grants DMS 9900543 and DMS 0200877