### Abstract

Let M be a locally symmetric irreducible closed manifold of dimension ≥3. A result of Borel 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 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.

Original language | English (US) |
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Journal | Advances in Mathematics |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

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### Keywords

- Locally symmetric spaces
- Manifolds
- Rigidity
- Transformation groups

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*. https://doi.org/10.1016/j.aim.2017.06.010

**A trichotomy theorem for transformation groups of locally symmetric manifolds and topological rigidity.** / Cappell, Sylvain; Lubotzky, Alexander; Weinberger, Shmuel.

Research output: Contribution to journal › Article

*Advances in Mathematics*. https://doi.org/10.1016/j.aim.2017.06.010

}

TY - JOUR

T1 - A trichotomy theorem for transformation groups of locally symmetric manifolds and topological rigidity

AU - Cappell, Sylvain

AU - Lubotzky, Alexander

AU - Weinberger, Shmuel

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Let M be a locally symmetric irreducible closed manifold of dimension ≥3. A result of Borel 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 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.

AB - Let M be a locally symmetric irreducible closed manifold of dimension ≥3. A result of Borel 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 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.

KW - Locally symmetric spaces

KW - Manifolds

KW - Rigidity

KW - Transformation groups

UR - http://www.scopus.com/inward/record.url?scp=85040567788&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85040567788&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2017.06.010

DO - 10.1016/j.aim.2017.06.010

M3 - Article

AN - SCOPUS:85040567788

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -