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The Kashiwara-Vergne (KV) conjecture is a property of the Campbell-Hausdorffseries put forward in 1978. It has been settled in the positive by E.Meinrenken and the first author in 2006. In this paper, we study the uniquenessissue for the KV problem. To this end, we introduce a family of infinitedimensional groups KV_n, and an extension \hat{KV}_2 of the group KV_2. We showthat the group \hat{KV}_2 contains the Grothendieck-Teichmueller group GRT as asubgroup, and that it acts freely and transitively on the set of solutions ofthe KV problem Sol(KV). Furthermore, we prove that Sol(KV) is isomorphic to adirect product of a line \k (\k being a field of characteristic zero) and theset of solutions of the pentagon equation with values in the group KV_3. Thelatter contains the set of Drinfeld's associators as a subset. As a by-product,we obtain a new proof of the Kashiwara-Vergne conjecture based on theDrinfeld's theorem on existence of associators.

The Kashiwara-Vergne conjecture and Drinfeld's associators