Margulis lemma for compact lie groups

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 \(\Gamma\subset G\) , \(U\cap \Gamma\) 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/H0, has an abelian subgroup of index \(\le w(n)\) , where H0 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 \(\le w(n)\) . We show this same property for the fundamental groups of almost non-negatively curved n-manifolds whose universal coverings are not collapsed.