Translation surfaces and the curve graph in genus two

Abstract

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $\left(X,\omega \right)\in \Omega {\mathcal{ℳ}}_2=\mathcal{\mathscr{H}}\left(2\right)\bigsqcup \mathcal{\mathscr{H}}\left(1,1\right)$ a subgraph ${\widehat{\mathcal{C}}}_{cyl}$ 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 ${\widehat{\mathcal{C}}}_{cyl}$ is by definition ${GL}^{+}\left(2,ℝ\right)$–invariant. Hence it may be seen as the image of the corresponding Teichmüller disk in the curve graph. We will show that ${\widehat{\mathcal{C}}}_{cyl}$ is always connected and has infinite diameter. The group ${Aff}^{+}\left(X,\omega \right)$ of affine automorphisms of $\left(X,\omega \right)$ preserves naturally ${\widehat{\mathcal{C}}}_{cyl}$, we show that ${Aff}^{+}\left(X,\omega \right)$ is precisely the stabilizer of ${\widehat{\mathcal{C}}}_{cyl}$ in $Mod\left(S\right)$. We also prove that ${\widehat{\mathcal{C}}}_{cyl}$ is Gromov-hyperbolic if $\left(X,\omega \right)$ is completely periodic in the sense of Calta.

It turns out that the quotient of ${\widehat{\mathcal{C}}}_{cyl}$ by ${Aff}^{+}\left(X,\omega \right)$ is closely related to McMullen’s prototypes in the case that $\left(X,\omega \right)$ is a Veech surface in $\mathcal{\mathscr{H}}\left(2\right)$. We finally show that this quotient graph has finitely many vertices if and only if $\left(X,\omega \right)$ is a Veech surface for $\left(X,\omega \right)$ in both strata $\mathcal{\mathscr{H}}\left(2\right)$ and $\mathcal{\mathscr{H}}\left(1,1\right)$.