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We study three graph complexes related to the higher genus Grothendieck–Teichmüller 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.

We establish an isomorphism between the Grothendieck-Teichmüller Lie algebra grt 1 in depth two modulo higher depth and the cohomology of the two-loop part of the graph complex of internally connected graphs ICG(1). In particular, we recover all linear relations satisfied by the brackets of the conjectural generators σ 2k+1 modulo depth three by considering relations among two-loop graphs.The Grothendieck-Teichmüller Lie algebra is related to the zeroth cohomology of M. Kontsevich's graph complex GC 2 via T. Willwacher's isomorphism. We define a descending filtration on H 0 (GC 2 ) and show that the degree two components of the corresponding associated graded vector spaces are isomorphic under T. Willwacher's map.

It is conjectured that the Kashiwara-Vergne Lie algebra krv 2 is isomorphic to the direct sum of the Grothendieck-Teichmüller Lie algebra grt 1 and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of krv 2 whose intersection is grt 1 , thus giving a way to interpolate between these two Lie algebras.2010 Mathematics Subject Classification. 17B65, 81R99.