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In this paper we show that, after completing in the -adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve with a quasi-algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper on the Goldman bracket to construct torsors of solutions to the Kashiwara–Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.
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Richard Hain, Hodge theory of the Turaev cobracket and the Kashiwara–Vergne problem. J. Eur. Math. Soc. 23 (2021), no. 12, pp. 3889–3933DOI 10.4171/JEMS/1088