A topological characterisation of the Kashiwara–Vergne groups

Abstract:

In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to the Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in $\mathbb {R}^4$, which can be finitely presented algebraically as a circuit algebra, or equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups $\mathsf {KV}$ and $\mathsf {KRV}$—the symmetry groups of Kashiwara-Vergne solutions—as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebras of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmüller group $\mathsf {GRT}_1$ as automorphisms of arrow diagrams.