  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.
