About the even minimal stratum of translation surfaces in genus 4

Abstract

In the present note, we complete the correspondence between stratum components of translation surfaces in low genus and finite-type Artin groups with defining Dynkin diagram containing $E_6$. In an earlier work, we showed that in genus $3$ the monodromy of the non-hyperelliptic connected components $\mathcal{H}^{\operatorname{odd}}(4)$ and $\mathcal{H}(3,1)$ are highly non-injective, as the respective kernels contain a non-abelian free group of rank $2$. The result holds since both the stratum components are orbifold classifying spaces for central extensions of the inner automorphism groups of the finite-type Artin groups $A_{E_6}$ and $A_{E_7}$, respectively. The following is a note extending the same result to the stratum $\mathcal{H}^{\operatorname{even}}(6)$ in genus $4$, which is an orbifold classifying space for a central extension of the group $\operatorname{Inn}(A_{E_8})$.