Graph Complexes and higher genus Grothendieck-Teichm\"uller Lie algebras

TL;DR

This paper presents a detailed description of the degree zero cohomology of graph complexes related to surfaces of any genus, revealing new higher genus Grothendieck-Teichmüller Lie algebras and their properties.

Contribution

It introduces a presentation of the cohomology of graph complexes for all genera, generalizing known structures and establishing new higher genus Grothendieck-Teichmüller Lie algebras.

Findings

Cohomology in degree zero is described by generators and relations.

For genus 1, recovers Enriquez's elliptic Grothendieck-Teichmüller Lie algebra.

Graph cohomology vanishes in negative degrees.

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-Teichm\"uller Lie algebra. In analogy to Willwacher's theorem relating Kontsevich's graph complex to Drinfeld's Grothendieck-Teichm\"uller Lie algebra, we call the results higher genus Grothendieck-Teichm\"uller Lie algebras. Moreover, we find that the graph cohomology vanishes in negative degrees.