Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$

Question

Looking at the images below, you recognize that the adjacency matrix of the graph $A_G$ splits up into three different colored submatrices, with $A_G=A_r+A_b+A_d$ (where $d$ is dark, damn...). It's obvious that $A_k^2=1$. Now have a look a the right coloring: You'll see that $(A_dA_r)^2=(A_rA_b)^2=(A_bA_d)^2=1$ as well. Let's use $a,b,c$ instead. We summarize this as: $$ \langle a,b,c|a^2=b^2=c^2=(ab)^2=(bc)^2=(ca)^2=1\rangle =\Delta(2,2,2) $$ This is a presentation of a triangle group $\Delta(2,2,2)$, a special kind of Coxeter group. The left image resembles the situation:

Now the right images deviates from the left at the inner square, where blue and dark are flipped; still a valid 3-edge coloring. It has Hamiltonian cycles (follow dark-red or red-blue edges), therefore its presentation is: $$ \langle a,b,c|a^2=b^2=c^2=(ab)^4=(bc)^4=(ca)^2=1 \rangle =\Delta(4,4,2) $$

So by flipping the inner cycle, we map $\Delta(2,2,2)\mapsto \Delta(4,4,2)$.

EDIT: This means that we change the classification from $\frac12+\frac12+\frac12=\frac32>1$, i.e. spherical, to $\frac14+\frac14+\frac12=1$, i.e. Euclidean.

Which branch of math deals with this mapping? What are good introductions to it?

Let's restrict to planar, bicubic graphs...

Answer 1

The branch of mathematics dealing with groups given by generators and relations, including Coxeter groups, is combinatorial group theory, which has developed into geometric group theory. One of the first monographs, still worth reading, is the book by Coxeter and Moser (the first author is the same man after whom Coxeter groups are named). Other classics are texts by Lyndon and Schupp and Magnus, Karras and Solitar. Many good modern introductions are available. You can search for italicized terms on MathSciNet or google, or post a streamlined question here on MO.