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We investigate periodic diffeomorphisms of non-compact aspherical manifolds(and orbifolds) and describe a class of spaces that have no homotopicallytrivial periodic diffeomorphisms. Prominent examples are moduli spaces ofcurves and aspherical locally symmetric spaces with non-vanishing Eulercharacteristic. In the irreducible locally symmetric case, we show that nocomplete metric has more symmetry than the locally symmetric metric. In themoduli space case, we build on work of Farb and Weinberger and prove ananalogue of Royden's theorem for complete finite volume metrics.

Smith theory, L2 cohomology, isometries of locally symmetric manifolds and moduli spaces of curves