## Abstract

We improve Margulis lemma for a compact connected Lie group G: there is a neighborhood U of the identity such that for any finite subgroup Γ ⊂ G, U ∩ Γ generates an abelian group. We show that for each n, there exists an integer w(n) > 0, such that if H is a closed subgroup of a compact connected Lie group G of dimension n, then the quotient group, H/H _{0}, has an abelian subgroup of index ≤ w(n) , where H _{0} is the identity component of H. As an application, we show that the fundamental group of the homogeneous space G/H has an abelian subgroup of index ≤ w(n). We show this same property for the fundamental groups of almost non-negatively curved n-manifolds whose universal coverings are not collapsed.

Original language | English (US) |
---|---|

Pages (from-to) | 395-406 |

Number of pages | 12 |

Journal | Mathematische Zeitschrift |

Volume | 258 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2008 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)