Purpose of imbedding a group onto a surface?

Question

I'm reading the book "Topological Graph Theory" by Gross and I've gone through a fair bit of it. It seems like the entire book is leading up to being able to imbed a group onto a surface, and I have no idea why you would want to do that. I am a physics major and not very advanced in math. Any insight would be appreciated!

Answer 1

The embeddings one generally encounters first are planar embeddings. A good example is the three cottage problem:

Suppose there are three cottages on a plane (or sphere) and each needs to be connected to the water, gas, and electricity companies. Without using a third dimension or sending any of the connections through another company or cottage, is there a way to make all nine connections without any of the lines crossing each other?

You could also consider any other planar embedding problem, except for on another surface, instead. What if your 3 cottages were on a sphere? (Perhaps a tiny planet, if you'll excuse the reference?) What if you wanted to print a circuit board on a torus, and wanted to be sure there were no short circuits due to edge intersections?

Answer 2

Many graph theoretic problems become easy when it is known that the graph is planar ie. can be embedded in the sphere. For example the isomorphism problem for planar graphs can be solved in polynomial time, they can be 5colored in polynomial time (I don’t know about the time complexity of finding a 4coloring though) etc. I would guess that the complexity of these problems increases with the complexity of the (in some sense minimal) surfaces you can embedd into. I would bet that this has been studied extensively, but I don’t know a reference. Moreover I bet that one can classify graphs by the surfaces they embedd into.