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