Using the intersection and self-intersection of loops on a surface one can define the Goldman-Turaev Lie bialgebra, and its non-commutative double avatar. On a genus zero surface with three boundary components the linearization problem of this structure is equivalent to the Kashiwara-Vergne problem in Lie theory. Motivated by this result a generalization of the Kashiwara-Vergne problem in higher genera is proposed and solutions are constructed in analogy with elliptic associators.

This is joint work with A. Alekseev, N. Kawazumi and Y. Kuno.