A trichotomy theorem for transformation groups of locally symmetric manifolds and topological rigidity

Let M be a locally symmetric irreducible closed manifold of dimension ≥3. A result of Borel [6] combined with Mostow rigidity imply that there exists a finite group G=G(M) such that any finite subgroup of Homeo+(M) is isomorphic to a subgroup of G. Borel [6] asked if there exist M's with G(M) trivial and if the number of conjugacy classes of finite subgroups of Homeo+(M) is finite. We answer both questions: