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Is there a general study of the symmetries of tilings on surfaces? Conway, Goodman-Strauss & Burgiel classified them on $\mathbb S^2, \mathbb R^2$ and $\mathbb H^2$, with their 'Magic Theorem'. But I found nothing about general surfaces, except for the cylinder which is an easy consequence from the result above.

I thought W. Thurston could have some results about this, but I found nothing.

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As the OP points out, it is natural to study tilings from the viewpoint of symmetries acting on a space. From this perspecitve, finite subgroups of $SO(3)$ lead to tilings of $\mathbb{S}^2$ and the wallpaper groups yield tilings of $\mathbb{R}^2$.

Along the same lines, Chapter 7 of Farb, Benson, and Dan Margalit. A Primer on Mapping Class Groups (PMS-49). Princeton University Press, 2011, provides a very nice treatment of the problem for surfaces of genus $g\geq 2$ using the mapping class group.

First, they discuss the $84(g − 1)$ theorem, aka Hurwitz's automorphisms theorem, which shows that the orientation preserving symmetry group of $S_g$ is at most $84(g-1)$. Using (orientable) covers of the orbifold quotient of the $(2,3,7)$ triangle group one can see this bound is sharp.

Also, Chapter 13 of Thurston's notes provides a good bit of background.

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John Conway The Symmetries of Things discusses the Magic Theorem in detail as well as versions with 2 or more colors. Echoing Neil's response, the quotient of $S^2$, $\mathbb{R}^2$ or $\mathbb{H}^2$ by a group action is going to be a Riemann surface of some kind. By uniformaztiions any equation like $y^2 = x(x-1)(x+1)$ is going to be a surface of this kind.

Beyond that I would like to know which surface? The Cylinder $S^1 \times \mathbb{R}$ has the Euclidean plane as a covering space, so Conway might feel it has been covered in his book.

The "things" which Conway discusses symmetry might not be spaces at all. The might be spaces of Modular forms (which live on the surfaces).

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  • $\begingroup$ I was thinking about general Riemann surfaces. $\endgroup$ – Melquíades Ochoa Apr 16 '16 at 0:46
  • $\begingroup$ @ArcadioBuendía how to general Riemann surfaces have symmetry $\endgroup$ – john mangual Apr 16 '16 at 13:37

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