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Pietro, for an introduction to this area written by combinatorics people I suggest you have a look at the paper "Introduction to Cohen-Macaulay posets" by Bjorner, Garsia and Stanley. The reasons to study the group action on top homology of a CM poset are, for example, briefly discussed in Sec. 6b of that paper.
Keith, thanks! Why don't you post it as an answer here? I think it's a bit misleading that the accepted answer to this question states that every nxn traceless matrix over a PID is a commutator!
@Kevin: this confirms my feeling - this paper does not claim it for matrices bigger than 2x2. Moreover, on the second page of the article, they claim explicitly that they do not know the answer for the 3x3 case!
@Wadim: thanks, this all sounds very interesting! As it often happens on MathOverflow, I did not expect to learn so many cool new things when asking such a naive question.
