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: (1) For every finite group G there exist M's with G(M)=G, and(2) the number of maximal subgroups of Homeo⁺(M) can be either one, countably many or continuum and we determine (at least for dim⁡M≠4) when each case occurs.Our detailed analysis of (2) also gives a complete characterization of the topological local rigidity and topological strong rigidity (for dimM≠4) of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces.