---
tag: FZ798EZ6M0
kind: concept
name: Zassenhaus neighbourhood
msc: ["22E40", "22E15"]
sources:
  - url: https://en.wikipedia.org/wiki/Margulis_lemma
---

## Summary

A Zassenhaus neighbourhood of a [[Lie group|7JBRGCVJRB]] $G$ is a neighbourhood $U$ of the identity with the following property.
The elements of any [[discrete subgroup|5M7D1BSHVZ]] that lie in $U$ generate a [[nilpotent group|0Q7MMQHC1P]].
A theorem of Zassenhaus says that every Lie group has such a neighbourhood.
It is the algebraic heart of the [[Margulis lemma|P6WSG7DT4Q]] and of the [[Kazhdan–Margulis theorem|C34RZMKKA4]].
The reason is that commutators of elements near the identity are much closer to the identity than the elements themselves.

## Explanations

### L1 human

In a Lie group, two elements close to the identity almost commute: their commutator is closer still to the identity.
In a discrete subgroup, elements cannot get arbitrarily close to the identity.
So taking commutators of small elements again and again must end at the identity after finitely many steps.
That is what being nilpotent means, and a neighbourhood where this argument works is a Zassenhaus neighbourhood.
In an abelian group like $\R^n$, every neighbourhood works, since all subgroups are abelian.