The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera

Abstract

For a compact oriented surface $\Sigma$ of genus $g$ with $n+1$ boundary components, the space $\mathfrak{g}(\Sigma)$ spanned by free homotopy classes of loops in $\Sigma$ carries the structure of a Lie bialgebra equipped with a natural decreasing filtration, whose structure morphisms are called the Goldman bracket and the (framed) Turaev cobracket. We address the following Goldman-Turaev (GT) formality problem: construct a Lie bialgebra homomorphism $\theta$ from $\mathfrak{g}(\Sigma)$ to its associated graded ${\rm gr}\, \mathfrak{g}(\Sigma)$ such that ${\rm gr} \, \theta = {\rm id}$. In order to solve it, we define a family of higher genus Kashiwara-Vergne (KV) problems for an element $F\in {\rm Aut}(L)$, where $L$ is a free Lie algebra. In the case of $g=0$ and $n=2$, it is the classical KV problem from Lie theory. For $g>0$, these KV problems are new. We show that an element $F$ induces a GT formality map if and only if it is a solution of the KV problem. A crucial step in solving the higher genus KV problem is to construct solutions for the case of $g=1$ and $n=1$ in terms of certain elliptic associators following Enriquez. By solving the KV problem, we establish the GT formality for every $g$ and $n$, with the exception of some framings for $g=1$ in which case the GT formality actually does not hold. Furthermore, we introduce pro-unipotent groups ${\rm KV}$ and ${\rm KRV}$ which act on the space of solutions of the KV problem freely and transitively. There are injective maps ${\rm GT}_1\to {\rm KV}, {\rm GRT}_1\to {\rm KRV}$ from Grothendieck-Teichm\"uller groups. As an application, we show that the Johnson obstruction given by the Turaev cobracket coincides with the one given by the Enomoto-Satoh trace. As part of our study, we prove a uniqueness theorem for non-commutative divergence cocycles on the group algebra of a free group which is of independent value.