Internally connected graphs and the Kashiwara-Vergne Lie algebra

It is conjectured that the Kashiwara-Vergne Lie algebra \(\widehat{\mathfrak {krv}}_2\) is isomorphic to the direct sum of the Grothendieck-Teichmüller Lie algebra \(\mathfrak {grt}_1\) and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of \(\widehat{\mathfrak {krv}}_2\) whose intersection is \(\mathfrak {grt}_1\), thus giving a way to interpolate between these two Lie algebras.