The Mal'cev Lie algebra associated to the Torelli group of a surface was completely determined by Hain (1993), who gave an explicit presentation which is quadratic if the genus is at least 4. I will explain some work, joint with A. Kupers, which exploits the formal similarity between surfaces and certain higher-dimensional manifolds to prove some new results about this Lie algebra: it is stably Koszul, and the geometric Johnson homomorphism is (nearly) stably injective.

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