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A functional equation arising in the theory of convergence groups. (English) Zbl 0826.39006
The main result contained in this paper is the following. Suppose the map \(\psi : \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) is continuous in one variable, locally bounded, continuous at \((0, 0)\), and satisfies (i) \(\psi (u,s) + \psi (v,u + s) = \psi (v,s) + \psi (u,v + s)\) and (ii) \(\psi (u,0) = 0\). Then there is a map \(Q : \mathbb{R} \to \mathbb{R}\) which is continuous and satisfies (a) \(\psi (u,v) = Q(u + v) - Q(u) - Q(v)\) and (b) \(Q(0) = 0\). Together, (i) and (ii) imply that \(\psi\) is a symmetric cocycle. Such functions have been shown to be of the form (a) by several previous authors, as cited in the paper. (Condition (ii) together with (i) implies symmetry, and with (a) gives (b) immediately.)
The authors also point out that, with a result of de Bruijn, the one- variable continuity of \(\psi\) is sufficient to guarantee the existence of a continuous \(Q\). They declare that the extra boundedness and continuity (joint) assumptions arise naturally in the context of convergence groups, and that their proof proceeds very simply and naturally. The reviewer is not knowledgable on the first point and is not convinced of the second.

39B22 Functional equations for real functions
30C62 Quasiconformal mappings in the complex plane
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