About the even minimal stratum of translation surfaces in genus 4

TL;DR

The paper analyzes the non-hyperelliptic minimal stratum $\mathcal{H}^{\operatorname{even}}(6)$ in genus $g=4$ by connecting it to versal deformation spaces of plane curve singularities associated with the root system $E_8$. Using Pinkham's framework, it identifies $\mathbb{P}\mathcal{H}^{\operatorname{even}}(6)$ with the moduli of pointed curves $\mathcal{M}_{4,1}^{\Gamma}$ for $\Gamma=\langle3,5\rangle$, and shows this projective stratum is governed by the versal deformation space $U_{E_8}$, a $K(\pi,1)$ for the Artin group $A_{E_8}$. The main result is that the orbifold fundamental group $\pi_1^{orb}(\mathbb{P}\mathcal{H}^{\operatorname{even}}(6))$ is isomorphic to the inner automorphism group $\operatorname{Inn}(A_{E_8})$, making $\pi_1^{orb}(\mathcal{H}^{\operatorname{even}}(6))$ a central extension of this group. Crucially, the kernel of the associated monodromy map to the mapping class group $\operatorname{Mod}(\Sigma_{4,1})$ contains a non-abelian free group of rank $2$, illustrating a large, non-injective monodromy and revealing intricate Teichmüller stratum topology via group-theoretic methods rooted in $E_8$ geometry.

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 $\operatorname{Inn}(A_{E_8})$.

Figures (3)

• Figure 1: The cubic $C$ on the cone $Q$ with the tangent points $p,q_1,q_2$.
• Figure 2: An example of the path segments $\sigma$ (on the left of the picture) and of $\delta$ (on the right hand side of the picture) in the case $R=A_3$. The coloured area represents the complexified chamber $C+iC$.
• Figure 3: A correspondence between the $E_8$ Dinkin diagram on some closed curves on $\Sigma_{4,1}$. Each vertex corresponds to a simple closed curve on the punctured surface on the right-hand side. The geometric homomorphism sends each standard generator of $A_{E_8}$ to the corresponding Dehn twist.

Theorems & Definitions (25)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• Lemma 6
• proof : Proof of Proposition \ref{['gapstrata']}
• Lemma 7