Stable cohomology of graph complexes

TL;DR

This paper investigates the stable cohomology of certain graph complexes linked to the Grothendieck-Teichmüller Lie algebra and diffeomorphism groups, revealing new algebraic structures and solutions to the elliptic Kashiwara-Vergne problem.

Contribution

It computes the cohomology of these graph complexes in the stable limit and connects it to the Koszul property of the Malcev completion of the mapping class group, also linking elliptic associators to the Kashiwara-Vergne problem.

Findings

Cohomology of graph complexes stabilizes as genus g tends to infinity.

Malcev completion of the genus g mapping class group is Koszul in the stable limit.

Elliptic associators provide solutions to the elliptic Kashiwara-Vergne problem.

Abstract

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.