A stabilizer interpretation of the Grothendieck-Teichmüller group ${\mathsf{GRT}}_1(\mathbf{k})$

Abstract

If $\mathfrak{u}$ and $\mathfrak{v}$ are Lie algebras, then the product $\mathrm{Out}(\mathfrak{u})\times \mathrm{Out}(\mathfrak{v})$ of their outer automorphism groups naturally acts on the set of outer Lie algebra morphisms from $\mathfrak{u}$ to $\mathfrak{v}$; the stabilizer of the outer class of a given such morphism is then a subgroup of $\mathrm{Out}(\mathfrak{u})\times \mathrm{Out}(\mathfrak{v})$. We show that this leads to two related interpretation of the Grothendieck-Teichmüller group ${\mathsf{GRT}}_1(\mathbf{k})$, where $\mathfrak{u},\mathfrak{v}$ are the Lie algebra of infinitesimal braids on the plane (resp. framed infinitesimal braids on the sphere) with 3 and 4 (resp. 4 and 5) strands: namely, it can be expressed as the joint intersection of the stabilizer groups of the outer classes of certain strand doubling morphisms $\varphi$ and $\psi$ with ${\mathrm{Out}}^{\ast }(\mathfrak{u})\times \mathrm{Out}(\mathfrak{v})$, where ${\mathrm{Out}}^{\ast }(\mathfrak{u})$ is a subgroup of $\mathrm{Out}(\mathfrak{u})$ of outer classes of inertia-preserving automorphisms of $\mathfrak{u}$.

22 pages