Hyperelliptic translation surfaces and folded tori (Q386180)

Given a Riemann surface \(X\) and an orientation-preserving involution \(\sigma\) of \(X\), one considers the double cover \(\pi: \tilde{X} \rightarrow X\) with a lifted involution \(\tilde{\sigma}\) of \(\tilde{X}\). The surface \(\tilde{X}/ \tilde{\sigma}\) is called a twist of the surface \((X, \sigma)\). This paper is devoted to study torus twists, that is to say, twists \(\tilde{X} / \tilde{\sigma}\) of genus \(1\).NEWLINENEWLINEThe surface \(X\) is presented by edge-identified polygons in the complex plane. If all identifications are translations, the surface is called a translation surface, and otherwise a half-translation surface. If one selects a one-form on \(X\), it twists into a quadratic differential. When a quadratic differential is not the square of a one-form, it is called strict.NEWLINENEWLINEThe main result of the paper is that torus twists of a hyperelliptic translation surface are strict quadratic differentials. The work ends by listing all covers and quotients of surfaces of genus \(2\). The moduli space of translation surfaces of genus 2 is stratified in two strata, corresponding to surfaces with a single cone point of order \(2\), and to surfaces with two cone points of order 1. Surfaces in the first stratum are represented by a Swiss cross, and those in the second one by a regular decagon. All the possible covers in either case are listed in an Appendix.