Margulis lemma for compact lie groups

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 Gamma subset of G, U boolean AND Gamma 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.MathematicsSCI(E)0ARTICLE2395-40625