### Abstract

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 dimM≠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.

Original language | English (US) |
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Pages (from-to) | 25-46 |

Number of pages | 22 |

Journal | Advances in Mathematics |

Volume | 327 |

DOIs | |

State | Published - Mar 17 2018 |

### Keywords

- Locally symmetric spaces
- Manifolds
- Rigidity
- Transformation groups

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Advances in Mathematics*,

*327*, 25-46. https://doi.org/10.1016/j.aim.2017.06.010