Selecta Mathematica

, Volume 24, Issue 3, pp 2063–2092 | Cite as

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

  • Matteo Felder


We establish an isomorphism between the Grothendieck–Teichmüller 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–Teichmüller 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.


Grothendieck–Teichmüller Lie algebra Graph complexes 

Mathematics Subject Classification

17B65 81R10 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GenevaGeneva 4Switzerland

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