We consider a particular class of tessellations $\{p,q\}$ on a Poincaré disk. There are few examples where a regular graph for a particular tessellation has been obtained. It is done by identifying the vertices at the boundaries and then connecting them to generate a regular graph, which, in general, can be a non-trivial problem. For instance, for {3,7} tessellation, Klein obtained the 7-regular graph with 24 vertices and 84 edges as shown below- enter image description here

The compact Riemann surface of the above regular graph is a genus 3 surface. That just means the above regular graph can be embedded on a genus 3 surface. However, I am not interested in any kind of embedding.

Just to illustrate my point further. The next candidate for regular graph (or map) in {3,7} has a compact Riemann surface or Hurwitz surface with genus 7, also known as Macbeath surface. This can easily be shown for the dual graph of {3,7}, i.e., {7,3}. However, this can be extended to higher genus $g = 14, 17, 118,..$. Maybe the reason is that at these $g$, a regular map $M$ can be obtained because the automorphism group $A(M)$ contains two particular automorphisms: (1) Let us say $f$, which cyclic permutes the edges that are successive sides of one face of $M$; this is like a rotation. And (2), let us say $e$, which cyclically permutes the successive edges meeting at one vertex of this face; this is like a reflection of the face about that edge. These are the automorphism referred to as $\bf{R}$ and $\bf{S}$ in Coxeter's notation (from Generators and Relations for Discrete Groups). Lastly, as a verification, if we have a regular map, we correspondingly have a Petrie polygon of that map.

Is there a way to generate a regular map (or graph) for any tessellation $\{p,q\}$ on a Poincaré disk, even if for a particular genus?
If yes, is there an algorithm in the sense that there exist generators that can produce the whole regular map on applying to some set of vertices respecting the right edges for $\{p,q\}$ for a genus $g$?

My question is different from this question on the existence of regular tessellation on a closed surface $S$, which concerns mostly about the existence of such regular tessellation embedding on $S$. However, I am concerned about how can such regular tessellation be obtained?



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