LAUSR

We study three different topologies on the moduli space $$\mathcal {H}^\mathrm{loc}_m$$ 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 $$\mathcal {C}^{k,\alpha }$$ C k , α -topology, for some $$k>1$$ k > 1 , which do not admit any convergent subsequence in the pointed $$\mathcal {C}^{k+1}$$ C k + 1 -topology.