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- W4289667051 abstract "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 Teichmuller 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 $Gamma$." @default.
- W4289667051 created "2022-08-03" @default.
- W4289667051 creator A5049456687 @default.
- W4289667051 date "2018-08-28" @default.
- W4289667051 modified "2023-10-18" @default.
- W4289667051 title "Topological Veech dichotomy and tessellations of the hyperbolic plane" @default.
- W4289667051 doi "https://doi.org/10.48550/arxiv.1808.09329" @default.
- W4289667051 hasPublicationYear "2018" @default.
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