Related papers: Topological Veech dichotomy and tessellations of t…
Given a closed surface $S$ with finitely generated Veech group $G$ and its $\pi_1(S)$-extension $\Gamma$, there exists a hyperbolic space $\hat{E}$ on which $\Gamma$ acts isometrically and cocompactly. The space $\hat{E}$ is obtained by…
For each stratum of the space of translation surfaces, we introduce an infinite translation surface containing in an appropriate manner a copy of every translation surface of the stratum. Given a translation surface $(X, \omega)$ in the…
In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction either the flow in that direction is completely periodic or…
Given a lattice Veech group in the mapping class group of a closed surface $S$, this paper investigates the geometry of $\Gamma$, the associated $\pi_1S$--extension group. We prove that $\Gamma$ is the fundamental group of a bundle with a…
In this paper, we study the PSV construction, which provides a step by step method for obtaining tame translation surfaces with a suitable Veech group. In addition, we modify slightly this construction, and for each finitely generated…
We provide a complete classification of groups that can be realized as isometry groups of a translation surface $M$ with non-finitely generated fundamental group and no planar ends. Furthermore, we demonstrate that if $S$ has no…
Identifying parallel sides of a collection of Euclidean polygons yields a flat surface with cone points of angles multiples of 2 pi, naturally a compact Riemann surface but also an algebraic curve, and a hyperbolic surface. In general two…
We study infinite superelliptic curves as translation surfaces and explore their Veech groups. These objects are branched covering of the complex plane with branching over infinitely many points. We provide a criterion for isomorphism…
We describe a family of representations of $\pi_1(\Sigma)$ in PU(2,1), where $\Sigma$ is a hyperbolic Riemann surface with at least one deleted point. This family is obtained by a bending process associated to an ideal triangulation of…
Let $(M,\omega)$ be a translation surface such that every leaf of its horizontal foliation is either closed, or joins two zeros of $\omega$. Then, $M$ decomposes as a union of horizontal Euclidean cylinders. The $\textit{twist torus}$ of…
For an infinite type surface $\Sigma$, we consider the space of (marked) convex hyperbolic structures on $\Sigma$, denoted $H(\Sigma)$, with the Fenchel-Nielsen topology. The (big) mapping class group acts faithfully on this space allowing…
We study the Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms of a…
Flat surfaces that correspond to meromorphic $1$-forms or to meromorphic quadratic differentials containing poles of order two and higher are surfaces of infinite area. We classify groups that appear as Veech groups of translation surfaces…
In this paper we consider the symmetries of triangulable half-dilation structures on the sphere with four singularities. We show that all such surfaces can be produced by a tetrahedral construction. Using this construction, we calculate…
A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincar\'e disk. Consider the infinite-dimensional Teichm\"uller spaces of hyperbolic…
We study the SL(2,R)-infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the "no small virtual triangles" theorem of…
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichm\"uller space, right-angled Artin groups, and many other cubical groups. Given such a space $\mathcal X$, we build…
In this note we are interested in the dynamics of the linear flow on infinite periodic $\mathbb{Z}^d$-covers of Veech surfaces. An elementary remark allows us to show that the kernel of some natural representations of the Veech group acting…
We provide sharp bounds for the isoperimetric constants of infinite plane graphs (tessellations) with bounded vertex and face degrees. For example, if $G$ is a plane graph satisfying the inequalities $p_1 \leq \mbox{deg}\ v \leq p_2$ for $v…
We study and classify regular and semi-regular tessellations of Riemann surfaces of various genera and investigate their corresponding supersymmetric gauge theories. These tessellations are generalizations of brane tilings, or bipartite…