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

TL;DR

This paper constructs a genus four origami with an arithmetic Kontsevich--Zorich monodromy, revealing that large Veech groups do not necessarily imply non-arithmetic monodromy, and provides data on monodromy indices in genus 4.

Contribution

It demonstrates the existence of a genus four origami with an arithmetic monodromy, challenging assumptions about the relationship between Veech groups and monodromy groups.

Findings

Existence of a genus four origami with arithmetic monodromy

Veech group is as large as possible, equal to SL(2,Z)

Provides data on monodromy index and congruence level 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.