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

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.