An arithmetic Kontsevich--Zorich monodromy of a symmetric origami in genus 4

Abstract

We demonstrate the existence of a certain genus four origami whose Kontsevich--Zorich monodromy is arithmetic in the sense of Sarnak. The surface is interesting because its Veech group is as large as possible and given by $\mathrm{SL}(2,\mathbb Z)$. When compared to other surfaces with Veech group $\mathrm{SL}(2,\mathbb Z)$ such as the Eierlegendre Wollmichsau and the Ornithorynque, an arithmetic Kontsevich--Zorich monodromy is surprising and indicates that there is little relationship between the Veech group and monodromy group of origamis. Additionally, we record the index and congruence level in the ambient symplectic group which gives data on what can appear in genus 4.