Zassenhaus neighbourhood

Summary

A Zassenhaus neighbourhood of a Lie group $G$ is a neighbourhood $U$ of the identity with the following property. The elements of any discrete subgroup that lie in $U$ generate a nilpotent group. A theorem of Zassenhaus says that every Lie group has such a neighbourhood. It is the algebraic heart of the Margulis lemma and of the Kazhdan–Margulis theorem. The reason is that commutators of elements near the identity are much closer to the identity than the elements themselves.