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I've been told that the difficulty in the mapping class group mainly lies in the Torelli group. To what extent is that true ? And why so ?

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I've heard the claim above from a professor and it stuck with me. I'm not sure what to make of it really because "difficulty" was not well defined. If you have an interesting comment to add, thank you. – Jim B May 4 '10 at 4:07
You know, most professors actually like it when students ask them for more information about something they said. (Most people do too, I've heard.) It's certainly a lot more efficient than asking a bunch of other mathematicians to explain what s/he meant. – Pete L. Clark May 4 '10 at 4:53
Answer: that is not true. Reason: the MCG is too interesting to be difficult for a single, well understood reason. If you want to know other reasons why the MCG is interesting, that would make a reasonable community wiki question. Even better: search google for survey articles on the MCG. – Sam Nead May 4 '10 at 4:56
A suggestion for such a survey if you are interested in the Torelli group : "A Survey of the Torelli Group" by Dennis Johnson (the crazy genius behind much of the foundational work on the Torelli group). – Andy Putman May 4 '10 at 14:06

The Torelli group is the kernel of a surjective homomorphism to $Sp_{2g}(\mathbb{Z})$, which is a much simpler group than the mapping class group (since it's Lie theoretic in nature). If you want a better answer than that, you have to ask a more specific and well-thought out question.

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