This is very helpful. What about when you have punctures?

That's a very interesting looking thesis, and I'm glad you brought it to my attention. However neither it nor Korkmaz's nice survey have anything like what I want. Korkmaz tells me only first and second homology, even though there are lots of other unstable classes known, while the thesis doesn't even deal with rational coefficients.

Counterpoint is appreciated.

That one also annoys me.

"Reversal" is another noun with an "al" ending.

"Finite quotient" usually means that $G/\Gamma$ is finite not that $\Gamma$ is finite as in your question.

@Igor: I didn't interpret the question that way. I interpreted it to mean, what's true in two dimensions that's not true in general.

Wow! Never in a million years would I have guessed there was an explicit formula, and what a formula it is!

This duplicates a question at math.stackexchange. math.stackexchange.com/questions/45815/counting-graphs

@Zuriel: That's true.