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 R 4 \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 K V \mathsf {KV} and K R V \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 G R T 1 \mathsf {GRT}_1 as automorphisms of arrow diagrams.