Filtrations on graph complexes and the Grothendieck–Teichmüller Lie algebra in depth two

Abstract

We establish an isomorphism between the Grothendieck–Teichmuller Lie algebra $$\mathfrak {grt}_1$$ in depth two modulo higher depth and the cohomology of the two-loop part of the graph complex of internally connected graphs $$\mathsf {ICG}(1)$$ . In particular, we recover all linear relations satisfied by the brackets of the conjectural generators $$\sigma _{2k+1}$$ modulo depth three by considering relations among two-loop graphs. The Grothendieck–Teichmuller Lie algebra is related to the zeroth cohomology of Kontsevich’s graph complex $$\mathsf {GC}_2$$ via Willwacher’s isomorphism. We define a descending filtration on $$H^0(\mathsf {GC}_2)$$ and show that the degree two components of the corresponding associated graded vector spaces are isomorphic under Willwacher’s map.