Topological Veech dichotomy and tessellations of the hyperbolic plane

For every half-translation surface with marked points $(M,\Sigma)$, we construct an associated tessellation $\Pi(M,\Sigma)$ of the Poincar\'e upper half plane whose tiles have finitely many sides and area at most $\pi$. The tessellation $\Pi(M,\Sigma)$ is equivariant with respect to the action of $\mathrm{PSL}(2,\mathbb{R})$, and invariant with respect to (half-)translation covering. In the case $(M,\Sigma)$ is the torus $\mathbb{C}/\mathbb{Z}^2$ with a one marked point, $\Pi(\mathbb{C}/\mathbb{Z}^2,\{0\})$ 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 Teichm\"uller curve in the case $(M,\Sigma)$ is a Veech surface (lattice surface). Under the assumption that $(M,\Sigma)$ satisfies the topological Veech dichotomy, there is a natural graph $\mathcal{G}$ underlying $\Pi(M,\Sigma)$ on which the Veech group $\Gamma$ acts by automorphisms. We show that $\mathcal{G}$ has infinite diameter and is Gromov hyperbolic. Furthermore, the quotient $\overline{\mathcal{G}}:=\mathcal{G}/\Gamma$ is a finite graph if and only if $(M,\Sigma)$ 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 $\Gamma$.