Dilation surfaces and their Veech groups

Figure 1.  A translation surface of genus $ 2 $

Figure 2.  A 'dilation surface' of genus $ 2 $ and a leaf of its horizontal foliation

Figure 3.  A 'hyperbolic' closed leaf

Figure 5.  The Franco-Russian slit construction

Figure 4.  A Hopf torus and the basis of its homology

Figure 6.  The double-chamber surface

Figure 7.  Dilation cylinders of the double-chamber surface

Figure 8.  The disco surface $ \operatorname{D}_{a, b} $

Figure 9.  An alternative representation of the disco surface

Figure 10.  Cut-and-paste operation applied to the image of the double-chamber surface under the matrix $ \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} $

Figure 11.  A ribbon graph with two vertices

Figure 12.  A cylinder decomposition of the surface of genus $ 2 $

Figure 13.  A dilation torus, which is not a Hopf torus

Figure 14.  A dilation surface with a non-discrete set of holonomy vectors of saddle connections starting at the black point

Figure 15.  An angular section in which all leaves are hyperbolic

Figure 16.  Topological setting of the triangulation