Ribbon graph complexes, the Grothendieck-Teichmueller group and Goldman-Turaev Lie bialgebra

Using recent results of Chan-Galatius-Payne “Tropical curves, graph complexes, and top weight cohomology of $cM_g$” (preprint arXiv:1805.10186) it is not hard to show that the Grothendieck-Teichmueller Lie algebra grt injects into the cohomology Lie algebra of the ribbon graph complex introduced in the paper of Merkulov-Willwacher “Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves”  (preprint  arXiv:1511.07808) in the context of a study of the totality of cohomology groups  of moduli spaces of algebraic curves with (skewsymmetrized) punctures.

In my talk I explain that the latter ribbon graph complex controls also universal deformations of the Goldman-Turaev Lie bialgebra structure on the free loop space of a genus g Rieman surface with n+1 boundaries. Hence every element of grt
gives us a universal and highly non-trivial deformation of the  latter.

The construction can be made rather transparent.  We show explicitly the first non-trivial contribution to the standard
Goldman-Turaev bracket and co-bracket coming from Kontsevich’s tetrahedron class in grt.

The talk is based on a work in progress.