Convergence of locally homogeneous spaces

We study three different topologies on the moduli space H-m(loc) of equivariant local isometry classes of m-dimensional locally homogeneous Riemannian spaces. As an application, we provide the first examples of locally homogeneous spaces converging to a limit space in the pointed C-k,C-alpha-topology, for some k > 1, which do not admit any convergent subsequence in the pointed Ck+1-topology.