  We study three graph complexes related to the higher genus
Grothendieck-Teichm\"uller Lie algebra and diffeomorphism groups of manifolds.
We show how the cohomology of these graph complexes is related, and we compute
the cohomology as the genus $g$ tends to $\infty$. As a byproduct, we find that
the Malcev completion of the genus $g$ mapping class group relative to the
symplectic group is Koszul in the stable limit (partially answering a question
of Hain). Moreover, we obtain that any elliptic associator gives a solution to
the elliptic Kashiwara-Vergne problem.
