Mathematics Colloquia and Seminars

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The mapping class group Mod(S) of a genus g surface S=S_g is the group of connected components of Homeo(S). The action of Mod(S) on H_1(S,Z) gives a surjective homomorphism from Mod(S) to Sp(2g,Z). The Torelli group I(S) is the kernel, and it thus encapsulates the nonarithmetic behavior of Mod(S). In this talk, we will explain both classical and new results on the finiteness properties of I(S). For example, Johnson showed that I(S_g) is finitely generated for g at least 3. In joint work with Mladen Bestvina and Kai-Uwe Bux, we show that the cohomological dimension of I(S_g) is equal to 3g-5. We also show that H_{3g-5}(I(S_g)) is infinitely generated. In particular, these theorems give a new perspective on the celebrated theorem of Mess that I(S_2) is an infinitely generated free group. Our main tool is a new contractible complex, the "complex of minimizing cycles," on which I(S) acts.

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