A note on discrete uniform subgroups of Lie groups (Q1078682)

Let G be an n-dimensional connected Lie group with a left invariant Riemannian metric \(\rho\) on G. Let \(\| \|\) denote the norm on the Lie algebra \({\mathfrak G}\) of G corresponding to \(\rho\) and let k(\(\rho)\) denote the norm of the Lie bracket i.e. \(k(\rho)=\max \{\| [X,Y]\|:\) X,Y\(\in {\mathfrak G}\), \(\| X\| \leq 1\), \(\| Y\| \leq 1\}\). A subset L of G is said to be d-dense (with respect to \(\rho)\), iff \(\bar B(\)a,d)\(\cap L\neq \emptyset\) for any closed ball \(\bar B(\)a,d) of radius d on G. The following two theorems are proved. Theorem 1. If there exists a discrete d-dense subgroup \(L\subset G\) with \(k(\rho)\cdot d\leq \epsilon (n)=0,34/(1+\sqrt{n})\), then G is nilpotent. Theorem 2. If \(\rho\) is bi-invariant and there exists a discrete d-dense subgroup \(L\subset G\) with \(k(\rho)\cdot d\leq \delta (n)=0,95/(1+\sqrt{n})\) then G is abelian. Proofs are based on estimates for Zassenhaus neighbourhoods.