Consider the regular tiling $(m,n)$ in which $m$ $n$-agons meet at each vertex. Most of the time this tilings have to "live" in the hyperbolic plane. The edges of its polygons define a graph where two vertices are adjacent if they share an edge of a polygon. I would like to prove that most of these graphs have the infinite binary tree as a subgraph.
Of course, it would be impossible to show this for the regular tiling $(4,4)$ because the amount of vertices at a given distance from a given vertex grows quadratically, not exponentially like in the binary tree. However, for most $(m,n)$ this growth is exponential and so the infinite binary tree could fit. By staring at images of these tilings this seems completely plausible, but I am having troubles formalizing it.
I have been able to prove that this indeed happens in $(4,5)$. I tried to use Cayley graph's arguments but got lost. So basically I have the following questions:
- How can I know if the graph I described for $(m,n)$ is the Cayley graph of a group? And if so, is it the Cayley graph of a well-known group?
- Is my Cayley argument too complicated and there is a simpler proof?
- Do you have any tips for drawing the $(m,n)$ tiling?
- Can you help me with this proof or point me to some theory that may help?
Thanks in advance! : )