Related papers: Veech Groups and Triangulations of Half-Dilation P…

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…

We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R)…

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…

Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They are the geometric framework to study suspensions of affine…

This paper is the first part in a 2 part study of an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this…

A group-theoretical approach to the construction of quasiperiodic tilings of a Euclidean plane, possessing five-fold symmetry, is applied. Of the infinitely many of variants of quasiperiodic partitions of the plane, possessing the dihedral…

Given a compact oriented triangulated $3$-manifold we find a non-trivial condition satisfied by certain labelings of the tetrahedra by elements of an arbitrary abelian group which we call angle structures. Smoothness of the manifold is used…

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…

Let $G$ be a triangulation of the sphere with vertex set $V$, such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined $A_W$ to be the…

We start by describing how ideal triangulations on a surface degenerate under pinching of a multicurve. We use this process to construct a homomorphism from the Ptolemy groupoid of a surface to that of a pinched surface which is natural…

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…

The main ob jective of this research is to find the different types of elliptic triangulations for planar discs and spheres. We begin in Chapter 1 with the mandatory introduction. In the second chapter we define and study the notion of a…

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…

We study the large scale geometry of the upper triangular subgroup of PSL(2,Z[1/n]), which arises naturally in a geometric context. We prove a quasi-isometry classification theorem and show that these groups are quasi-isometrically rigid…

In this paper, we determine the topology of the spaces of convex polyhedra inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit $2$-sphere. These spaces can be regarded as discretized…

Veech groups are an important tool to examine translation surfaces and related mathematical objects. Origamis, also known as square-tiled surfaces, form an interesting class of translation surfaces with finite index subgroups of SL(2,Z) as…

The regular polyhedra have the highest order of 3D symmetries and are exceptionally at- tractive templates for (self)-assembly using minimal types of building blocks, from nano-cages and virus capsids to large scale constructions like glass…

The Solomon-Tits theorem says that the poset of proper non-trivial subspaces of a finite-dimensional vector space has realisation equivalent to a wedge of spheres. In this paper we prove a variant of this result for collections of geodesic…

We classify the dihedral edge-to-edge tilings of the sphere by regular polygons and quadrilaterals with equal opposite edges (edge configuration xyxy).

We study the simplicial volume of manifolds obtained from Davis' reflection group trick, the goal being characterizing those having positive simplicial volume. In particular, we focus on checking whether manifolds in this class with nonzero…