---
tag: P6WSG7DT4Q
kind: statement
role: theorem
name: Margulis lemma
msc: [53C20, "22E40", 20F65]
uses: [7CBT4SM83Y, 1X42JAA349, 02HPP5VCZH, 0Q7MMQHC1P, 0DTXBQPA2S]
about: [0Q7MMQHC1P]
sources:
  - url: https://en.wikipedia.org/wiki/Margulis_lemma
same_as:
  wikidata: Q6760835
  wikipedia: Margulis lemma
---

## Statement

For every $n \ge 1$ there is a constant $\varepsilon_n > 0$ with the following property.
Let $X$ be a complete, simply connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with [[sectional curvature|1X42JAA349]] between $-1$ and $0$.
Let $\Gamma$ be a discrete group of [[isometries|02HPP5VCZH]] of $X$, and let $x \in X$.
Let $\Gamma_x$ be the subgroup generated by the elements $\gamma \in \Gamma$ with $d(x, \gamma x) < \varepsilon_n$.
Then $\Gamma_x$ is virtually nilpotent: it has a [[nilpotent|0Q7MMQHC1P]] subgroup of finite [[index|0DTXBQPA2S]].

## Remarks

The proof combines a [[Zassenhaus neighbourhood|FZ798EZ6M0]] in a suitable Lie group with a compactness argument.
The curvature bounds give uniform control of the isometries that move a point only a little, which is why $\varepsilon_n$ depends only on $n$.

Example: in the upper half-plane model of $\mathbb{H}^2$, let $\Gamma$ be generated by the parabolic isometry $\gamma(z) = z + 1$.
At $x = i y$, the distance $d(x, \gamma x) = 2 \operatorname{arsinh}(1/(2y))$ tends to $0$ as $y \to \infty$.
So for large $y$ the group $\Gamma_x$ is all of $\Gamma \cong \Z$, which is abelian, as the lemma allows.

The lower curvature bound matters.
Multiplying the metric of $\mathbb{H}^2$ by $1/c$ gives curvature $-c$ and divides all displacements by $\sqrt{c}$.
For $c$ large, the generators of the fundamental group of a closed hyperbolic surface then move a point less than $\varepsilon_2$.
This group is not virtually nilpotent, so the conclusion fails without the bound $-1$.

## Explanations

### L1 human

Consider a group of symmetries of a negatively curved space, such as the hyperbolic plane, that acts without accumulating.
Look only at the symmetries that move a given point by a very small amount.
The lemma says that these symmetries generate a group that is almost commutative, namely virtually nilpotent.
The threshold for small depends only on the dimension.

### L2 human

The main application is the thick-thin decomposition of a hyperbolic manifold $M = \mathbb{H}^n/\Gamma$.
The thin part, where the injectivity radius is less than $\varepsilon_n / 2$, consists of tubes around short closed geodesics and of cusps.
This structure underlies the study of volumes of hyperbolic manifolds and of their convergence.
A Lie group form, with lattices in place of $\Gamma$, is closely related to the [[Kazhdan–Margulis theorem|C34RZMKKA4]].