Graph Complexes and higher genus Grothendieck-Teichmüller Lie algebras

Abstract

We give a presentation in terms of generators and relations of the cohomology in degree zero of the Campos-Willwacher graph complexes associated to compact orientable surfaces of genus $g$. The results carry a natural Lie algebra structure, and for $g=1$ we recover Enriquez' elliptic Grothendieck-Teichmüller Lie algebra. In analogy to Willwacher's theorem relating Kontsevich's graph complex to Drinfeld's Grothendieck-Teichmüller Lie algebra, we call the results higher genus Grothendieck-Teichmüller Lie algebras. Moreover, we find that the graph cohomology vanishes in negative degrees.

arXiv comment76 pages

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