Translation surfaces and the curve graph in genus two

TL;DR

This paper studies a special subgraph of the curve graph associated with genus two translation surfaces, proving its connectivity, infinite diameter, hyperbolicity under certain conditions, and characterizing when its quotient is finite.

Contribution

It introduces a new invariant subgraph of the curve graph for genus two translation surfaces and analyzes its geometric and group-theoretic properties, linking it to Veech surfaces and McMullen's prototypes.

Findings

The subgraph is always connected and has infinite diameter.

The subgraph is Gromov-hyperbolic if the surface is completely periodic.

The quotient of the subgraph by the affine automorphism group is finite if and only if the surface is a Veech surface.

Abstract

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ of the curve graph of $S$. The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic, or by a concatenation of two parallel saddle connections (satisfying some additional properties) on $X$. The subgraph $\hat{\mathcal{C}}_{\rm cyl}$ is by definition $\mathrm{GL}^+(2,\mathbb{R})$-invariant. Hence, it may be seen as the image of the corresponding Teichm\"uller disk in the curve graph. We will show that $\hat{\mathcal{C}}_{\rm cyl}$ is always connected and has infinite diameter. The group ${\rm Aff}^+(X,\omega)$ of affine automorphisms of $(X,\omega)$ preserves naturally $\hat{\mathcal{C}}_{\rm cyl}$, we show that ${\rm Aff}^+(X,\omega)$ is precisely the stabilizer of $\hat{\mathcal{C}}_{\rm cyl}$ in ${\rm Mod}(S)$. We also prove that $\hat{\mathcal{C}}_{\rm cyl}$ is Gromov-hyperbolic if $(X,\omega)$ is completely periodic in the sense of Calta. It turns out that the quotient of $\hat{\mathcal{C}}_{\rm cyl}$ by ${\rm Aff}^+(X,\omega)$ is closely related to McMullen's prototypes in the case $(X,\omega)$ is a Veech surface in $\mathcal{H}(2)$. We finally show that this quotient graph has finitely many vertices if and only if $(X,\omega)$ is a Veech surface for $(X,\omega)$ in both strata $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$.