Given a sequence of metric spaces, each space equipped with an isometric group action, the convergence of the sequence can be considered in the equivariant Gromov-Hausdorff topology.

 

In this talk, relationships between the groups of isometries in a convergent sequence will be discussed. When are there homomorphisms from groups in the tail of the sequence to the limit group? When can these be injective or surjective? With geometric assumptions on the metric spaces, strong results can be obtained. As an application, it is shown that certain classes of Riemannian orbifolds defined by geometric or spectral quantities have finitely orbifold homeomorphism types.