---
tag: NS2057MFWP
kind: concept
name: Lattice in a locally compact group
msc: ["22D05", "22E40", "28C10"]
specializes: [5M7D1BSHVZ]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup)
same_as:
  wikidata: Q6497088
  wikipedia: Lattice (discrete subgroup)
---

## Summary

A lattice in a [[locally compact group|AHV05ZBCYW]] $G$ is a [[discrete subgroup|5M7D1BSHVZ]] $\Gamma$ of finite covolume.
That is, the quotient $G/\Gamma$ has finite volume for a $G$-invariant measure.
It is cocompact, or uniform, when $G/\Gamma$ is compact.
Lattices are discrete approximations of $G$; the integer lattice $\Z^n$ in $\R^n$ and the modular group $SL_2(\Z)$ in $SL_2(\R)$ are the guiding examples.
They link group theory with number theory, geometry and ergodic theory.

## Explanations

### L1 human

The integer points $\Z^2$ form a grid in the plane, and the grid squares tile the plane by shifts.
Each tile has area $1$, so the plane "folded up" by the grid, which is a torus, has finite area.
A lattice in a general group is the same idea: a discrete subgroup whose translates of one region of finite volume cover the group.
The group $SL_2(\Z)$ of integer matrices with determinant $1$ sits inside $SL_2(\R)$ in this way, even though the folded-up space is not compact.