---
tag: BW21VJW403
kind: definition
name: Lattice in a locally compact group
msc: ["22D05", "22E40", "28C10"]
defines: NS2057MFWP
uses: [AHV05ZBCYW, 5M7D1BSHVZ, VY0HXD3GC4, HZNEQQBCW4, CXG6JZGMCK, KNJA2NEZ9F]
sources:
  - url: https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup)
---

## Definition

Let $G$ be a [[locally compact group|AHV05ZBCYW]] and let $\Gamma$ be a [[discrete subgroup|5M7D1BSHVZ]] of $G$.
Let $G/\Gamma$ be the space of left cosets $g\Gamma$ with the quotient topology, on which $G$ acts by $h \cdot g\Gamma = hg\Gamma$.
The subgroup $\Gamma$ is a *lattice* in $G$ if there is a nonzero [[Radon measure|HZNEQQBCW4]] $\nu$ on $G/\Gamma$ such that:

1. $\nu$ is $G$-invariant: $\nu(hE) = \nu(E)$ for every $h \in G$ and every Borel set $E \subseteq G/\Gamma$;
2. $\nu$ is finite: $\nu(G/\Gamma) < \infty$.

A lattice $\Gamma$ is *uniform*, or *cocompact*, if $G/\Gamma$ is compact.

## Remarks

Example: $\Z^n$ in $(\R^n, +)$.
The quotient $\R^n/\Z^n$ is the torus, and the image of Lebesgue measure on $[0, 1)^n$ is a translation-invariant measure on it of total mass $1$.
Since the torus is compact, $\Z^n$ is a uniform lattice.

Non-example: $\Gamma = \Z \times \set{0}$ in $\R^2$ is discrete, but fails clause 2.
The quotient is the cylinder $(\R/\Z) \times \R$.
Every invariant Radon measure on it is a multiple of the product of the [[Haar measures|VY0HXD3GC4]] of the two factors.
The nonzero ones have infinite total mass.

If $G$ has a lattice, then $G$ is [[unimodular|CXG6JZGMCK]].
A measurable [[fundamental domain|KNJA2NEZ9F]] for the action of $\Gamma$ on $G$ by right multiplication has finite Haar measure exactly when $\Gamma$ is a lattice.
The group $SL_2(\Z)$ is a lattice in $SL_2(\R)$ that is not uniform.