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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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 wellthought out question. 

