---
tag: P52TCSHP7R
kind: definition
name: Zassenhaus neighbourhood
msc: ["22E40", "22E15"]
defines: FZ798EZ6M0
uses: [7JBRGCVJRB, 5M7D1BSHVZ, 0Q7MMQHC1P, 0ZCY0Z80B6]
sources:
  - url: https://en.wikipedia.org/wiki/Margulis_lemma
---

## Definition

Let $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$.
A *Zassenhaus neighbourhood* of $G$ is an open [[neighbourhood|0ZCY0Z80B6]] $U$ of $e$ with the following property.
For every [[discrete subgroup|5M7D1BSHVZ]] $\Gamma$ of $G$, the subgroup generated by $\Gamma \cap U$ is [[nilpotent|0Q7MMQHC1P]].

## Remarks

Every Lie group has a Zassenhaus neighbourhood, by a theorem of Zassenhaus.
One can moreover choose $U$ so that $\Gamma \cap U$ lies in a connected nilpotent Lie subgroup of $G$.

Example: in $G = \R^n$, every open neighbourhood of $0$ is a Zassenhaus neighbourhood.
Indeed every subgroup of $\R^n$ is abelian, hence nilpotent.

Non-example: $U = \mathrm{SO}(3)$ in $G = \mathrm{SO}(3)$.
The rotation group of the icosahedron is a finite, hence discrete, subgroup isomorphic to the alternating group $A_5$.
It lies in $U$ and is not nilpotent, since $A_5$ is simple and non-abelian.