The motivic Grothendieck–Teichmüller criterion for Kashiwara–Vergne groups

Assume $2g+n>1$. Let ${\widehat{{\mathcal U}}}_{g,n+{\vec{1}}}^{\mathbf{d}}$ and ${\mathcal{KRV}}_{g,n+{\vec{1}}}^{\mathbf{d}}$ be the prounipotent groups associated with a framed surface and its Kashiwara–Vergne solutions, let ${\mathcal K}$ be the prounipotent radical of $\pi_1({\mathsf{MTM}})$, and let ${\mathrm{GRT}}$ be the de Rham version of the Grothendieck–Teichmüller group. Let ${\overline{{\mathcal U}}}_{g,n+{\vec{1}}}^{\mathbf{d}}$ be the normal subgroup of ${\widehat{{\mathcal U}}}_{g,n+{\vec{1}}}^{\mathbf{d}}$.

Motivic Grothendieck–Teichmüller criterion. The inclusion

$${\widehat{{\mathcal U}}}_{g,n+{\vec{1}}}^{\mathbf{d}}\to {\mathcal{KRV}}_{g,n+{\vec{1}}}^{\mathbf{d}}$$

is an isomorphism if and only if the inclusion

$$\pi_1({\mathsf{MTM}})\to {\mathrm{GRT}}$$

is an isomorphism. In this case, there is a split extension

$$1\to {\overline{{\mathcal U}}}_{g,n+{\vec{1}}}^{\mathbf{d}}\to {\mathcal{KRV}}_{g,n+{\vec{1}}}^{\mathbf{d}}\to {\mathcal K}\to 1.$$

This links the completeness of the Hodge-theoretic Kashiwara–Vergne construction to the motivic Grothendieck–Teichmüller comparison. The supplied text gives no resolution status; the preceding theorem establishes related structural results assuming the stated motivic mixed Tate hypothesis.