I was wondering about the research on countable connected graphs on compact surfaces, in particular the flat torus. In my considerations, I started with the flat torus view by $\mathbb{T}=[0,1]^2/{\sim}$, where $\sim$ is the identification of opposite sides.
Draw lines cutting the square into $4$ equals squares and draw a vertex where two lines meet. This gives a Graph consisting of one vertex and 2 loops embedded in $\mathbb{T}$. Inductively, proceed with subdiving each square of the previous step into $4$ identical squares.
In the limit, this gives a countable graph $G$ embedded in $\mathbb{T}$. If I am not completely mistaken, $G$ is transitive. Now I want to count the cycles in $G$ starting from some (and therefore any) vertex $v\in G$ with length smaller then some $k\in \mathbb{N}$. It looks like this is equivalent to counting rooted cycles in $\mathbb{Z}^2$ but I do not manage to grasp the structure since the area of the squares shrinks to zero and so does the length of the edges after embedding.
Can I approach this as is done for the integer lattice via the dual graph and counting number of connected areas of squares with a fixed boundary length? Or am I missing some subtleties here? My intuition breaks down due to the strange geometry of the lengths/areas and finite length paths at some stage of the inductive construction which grow exonzntially to infinity when continuing the subdivisions.