Topological Veech dichotomy and tessellations of the hyperbolic plane

For every half-translation surface with marked points $(M,Σ)$, we construct an associated tessellation $Π(M,Σ)$ of the Poincaré upper half plane whose tiles have finitely many sides and area at most $π$. The tessellation $Π(M,Σ)$ is equivariant with respect to the action of $\mathrm{PSL}(2,\mathbb{R})$, and invariant with respect to (half-)translation covering. In the case $(M,Σ)$ is the torus $\mathbb{C}/\mathbb{Z}^2$ with a one marked point, $Π(\mathbb{C}/\mathbb{Z}^2,\{0\})$ coincides with the iso-Delaunay tessellation introduced by Veech as both tessellations give the Farey tessellation. As application, we obtain a bound on the volume of the corresponding Teichmüller curve in the case $(M,Σ)$ is a Veech surface (lattice surface). Under the assumption that $(M,Σ)$ satisfies the topological Veech dichotomy, there is a natural graph $\mathcal{G}$ underlying $Π(M,Σ)$ on which the Veech group $Γ$ acts by automorphisms. We show that $\mathcal{G}$ has infinite diameter and is Gromov hyperbolic. Furthermore, the quotient $\overline{\mathcal{G}}:=\mathcal{G}/Γ$ is a finite graph if and only if $(M,Σ)$ is actually a Veech surface, in which case we provide an algorithm to determine the graph $\overline{\mathcal{G}}$ explicitly. This algorithm also allows one to get a generating family and a "coarse" fundamental domain of the Veech group $Γ$.

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