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 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.