Topological Veech dichotomy and tessellations of the hyperbolic plane

Abstract

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 PSL(2, ℝ), and invariant with respect to (half-)translation covering. In the case (M, Σ) is the torus ℂ/ℤ² with a one marked point, Π(ℂ/ℤ², {0}) coincides with the iso-Delaunay tessellation introduced by Veech [25] (see also [1, 2]) 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 ${\cal G}$ underlying Π(M, Σ) on which the Veech group Γ acts by automorphisms. We show that ${\cal G}$ has infinite diameter and is Gromov hyperbolic. Furthermore, the quotient $\overline {\cal G}:= {\cal G}/\Gamma$ 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 $\overline {\cal G}$ explicitly. This algorithm also allows one to get a generating family and a “coarse” fundamental domain of the Veech group Γ.