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

TL;DR

The work develops a higher-genus GT formality framework by tying the Goldman–Turaev Lie bialgebra structure on surfaces with boundary to a generalized Kashiwara–Vergne problem. It shows that solving the KV problem for a surface with framing generates a GT formality map, with genus-zero cases linked to associators and higher-genus cases reducible to key genus-zero and elliptic-genus one instances (via elliptic associators). The authors construct pro-unipotent groups KV and KRV acting freely and transitively on KV solution spaces and relate GT formality to Grothendieck–Teichmüller structures, while applications include a Johnson obstruction equality with the Enomoto–Satoh trace and a non-commutative divergence cocycle uniqueness result. Overall, the paper provides a rigorous bridge between topological surface invariants and non-commutative algebraic formality, with broad implications for mapping class groups and Johnson-type filtrations.

Abstract

For a compact oriented surface $Σ$ of genus $g$ with $n+1$ boundary components, the space $\mathfrak{g}(Σ)$ spanned by free homotopy classes of loops in $Σ$ 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 $θ$ from $\mathfrak{g}(Σ)$ to its associated graded ${\rm gr}\, \mathfrak{g}(Σ)$ such that ${\rm gr} \, θ= {\rm id}$. 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-Teichmüller 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.

Figures (16)

• Figure 1: The Goldman bracket and Turaev cobracket.
• Figure 2: The Goldman bracket.
• Figure 3: The operation $\kappa$
• Figure 4: Inserting monogons
• Figure 5: Splitting $\alpha$ at a self-intersection

Theorems & Definitions (376)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Definition 1.10