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We study Lie group structures on groups of the form C^\infty(M,K)}, where Mis a non-compact smooth manifold and K is a, possibly infinite-dimensional, Liegroup. First we prove that there is at most one Lie group structure with Liealgebra C^\infty(M,k) for which the evaluation map is smooth. We then prove theexistence of such a structure if the universal cover of K is diffeomorphic to alocally convex space and if the image of the left logarithmic derivative in\Omega^1(M,k) is a smooth submanifold, the latter being the case in particularif M is one-dimensional. We also obtain analogs of these results for the groupO(M,K) of holomorphic maps on a complex manifold with values in a complex Liegroup. We show that there exists a natural Lie group structure on O(M,K) if Kis Banach and M is a non-compact complex curve with finitely generatedfundamental group.Comment: 39 page

Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds