The Grothendieck-Teichmueller Lie algebra and Brown's dihedral moduli spaces

We prove that the degree zero Hochschild-type cohomology of the homology operad of Francis Brown’s dihedral moduli spaces is equal to the Grothendieck-Teichmueller Lie algebra plus two classes. This significantly elucidates the (in part still conjectural) relation between the Grothendieck-Teichmueller Lie algebra and (motivic) multiple zeta values.