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

Sylvain Cappell, Alexander Lubotzky, Shmuel Weinberger

Research output: Contribution to journalArticle

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 languageEnglish (US)
JournalAdvances in Mathematics
DOIs
StateAccepted/In press - Jan 1 2018

Fingerprint

Transformation group
Rigidity
Subgroup
Finite Group
Theorem
Semisimple Lie Group
Maximal Subgroup
Conjugacy class
Symmetric Spaces
Trivial
Continuum
Isomorphic
Imply
Closed

Keywords

  • Locally symmetric spaces
  • Manifolds
  • Rigidity
  • Transformation groups

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Advances in Mathematics, 01.01.2018.

Research output: Contribution to journalArticle

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