For a compact oriented surface Σ of genus g with n+1 boundary
components, the space g spanned by free homotopy classes of loops
in Σ carries the structure of a Lie bialgebra. The Lie bracket was
defined by Goldman and it is canonical. The Lie cobracket was defined by Turaev
and it depends on the framing of Σ. The Lie bialgebra g has
a natural decreasing filtration such that both the Goldman bracket and the
Turaev cobracket have degree (−2).
In this paper, we address the following Goldman-Turaev formality problem:
construct a Lie bialgebra homomorphism θ from g to its
associated graded grg such that grθ=id. In
order to solve it, we define a family of higher genus Kashiwara-Vergne (KV)
problems for an element F∈Aut(L), where L is a free Lie algebra. In the
case of g=0 and n=2, the problem for F is the classical KV problem from
Lie theory. For g>0, these KV problems are new.
Our main results are as follows. On the one hand, every solution of the KV
problem induces a GT formality map. On the other hand, higher genus KV problems
admit solutions for any g and n. In fact, the solution reduces to two
important cases: g=0,n=2 which admits solutions by Alekseev and Torossian
and g=1,n=1 for which we construct solutions in terms of certain elliptic
associators following Enriquez. By combining these two results, we obtain a
proof of the GT formality for any g and n.
We also study the set of solutions of higher genus KV problems and introduce
pro-unipotent groups KRV(g,n+1) which act on them freely and transitively.
These groups admit graded pro-nilpotent Lie algebras krv(g,n+1). We show
that the elliptic Lie algebra krv(1,1) contains a copy of the
Grothendieck-Teichmuller Lie algebra grt1​ as well as symplectic derivations
δ2n​.Comment: 61 pages, 5 figure