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We give a deterministic distributed approximation algorithm for the maximum matching problem in graphs of bounded arboricity. Specifically, given 0 驴a, the algorithm finds a matching of size at least (1 驴 驴)m(G), where m(G) is the size of the maximum matching in an n-vertex graph G with arboricity at most a. The algorithm runs in O(log* n) rounds in the message passing model and it is the first sublogarithmic algorithm for the problem on such a wide class of graphs. Moreover, the result implies that the known $\Omega(\sqrt{\log n/\log\log n})$ lower bound on the time complexity for a constant or polylogarithmic approximation does not hold for graphs of bounded arboricity.