---
tag: BTYF5Y5PFY
kind: concept
name: Lattice in a Lie group
msc: ["22E40"]
specializes: [NS2057MFWP]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup)
---

## Summary

A lattice in a [[Lie group|7JBRGCVJRB]] $G$ is a [[discrete subgroup|5M7D1BSHVZ]] $\Gamma$ such that the quotient $G/\Gamma$ has finite volume for a $G$-invariant measure.
It is the case of Lie groups of a [[lattice in a locally compact group|NS2057MFWP]].
The lattices in $\R^n$ are the classical [[lattices|WN9V7WSJKF]] of the geometry of numbers, and $\mathrm{SL}_n(\Z)$ is a lattice in $\mathrm{SL}_n(\R)$.
Lattices link Lie groups with number theory, through [[arithmetic groups|4XWG6S7R4Z]], and with geometry, as fundamental groups of finite-volume locally symmetric spaces.

## Explanations

### L1 human

A lattice is a discrete set of symmetries of a continuous group that is large enough to fill the group up to a piece of finite size.
The integer points $\Z^2$ in the plane are the model: they are spread out, yet one unit square of the plane already contains a copy of every point modulo $\Z^2$.
In a non-commutative group such as $\mathrm{SL}_2(\R)$, the integer matrices $\mathrm{SL}_2(\Z)$ play the same part.
Here the leftover piece is not compact, but it still has finite volume.