Outer space, symplectic derivations of free Lie algebras and modular forms

In this talk I will describe the connection, discovered by Kontsevich, between symplectic derivations of a free Lie algebra and the “symmetric space” for the group Out(F_n) of outer automorphisms of a free group. The latter is known as Outer space, and can be described as a space of free actions of F_n on metric simplicial trees. The fact that the quotients of such actions are finite graphs leads to a combinatorial understanding of this space which can be used to gain cohomological information about both the group Out(F_n) and the Lie algebra of symplectic derivations. One surprising outcome is a way of constructing cohomology classes from classical modular forms, as described in joint work with Conant and Kassabov. No prior knowledge of Outer space or Kontsevich’s theorem will be assumed

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