I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we need to move to the hyperbolic plane in order to obtain a tessellation representation, which makes sense because of Gauss-Bonnet's restriction that the total curvature of a surface of genus > 1 be negative.

But, since we had to move to the hyperbolic plane in order to obtain a tessellation, my question is this: can we only find tessellation representations of compact surfaces of genus > 1 when we impose constant negative curvature on them? Or does the tessellation representation look the same when I change curvature on certain sets? Or maybe, is it slightly perturbed but still a tessellation? If my questions are completely misguided, could someone suggest some literature? Thank you!


It is always easier to think about the spherical case first. You can see that all round, two-dimensional spheres, regardless of radius, admit tessellations. So constant curvature $+1$ is not necessary. On the other hand, constant positive curvature should be a requirement. (In fact, it is fair to require that the metric be homogeneous and isotropic. Otherwise what does it mean for tiles of the tessellation to be identical?)

The case of constant negative curvature is the same. The choice of constant doesn't really matter, so you might as well use $-1$. For an elementary discussion, with many beautiful pictures, see "Noneuclidean tesselations and their groups", by Wilhelm Magnus. A more modern treatment, also with wonderful graphics is "Indra's Pearls" by Mumford, Series, and Wright.

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