Published on *Max Planck Institute for Mathematics* (https://www.mpim-bonn.mpg.de)

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- Talk [1]

Speaker:

Sergei Merkulov
Affiliation:

Universität Luxemburg/MPIM
Date:

Mon, 2019-01-28 13:00 - 14:00 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.

**Links:**

[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] https://www.mpim-bonn.mpg.de/node/3444

[3] https://www.mpim-bonn.mpg.de/node/5312