---
tag: 4XWG6S7R4Z
kind: concept
name: Arithmetic group
msc: [20G30, 11F06, "22E40"]
sources:
  - url: https://en.wikipedia.org/wiki/Arithmetic_group
same_as:
  wikidata: Q4791126
  wikipedia: Arithmetic group
---

## Summary

An arithmetic group is a group of integer matrices cut out by polynomial equations, such as $\mathrm{SL}_n(\Z)$.
A group that agrees with such a group up to finite index is also arithmetic.
Arithmetic groups link number theory with the geometry of Lie groups.
For a semisimple group, a theorem of Borel and Harish-Chandra says that the arithmetic subgroups are lattices in the real Lie group.
The Margulis arithmeticity theorem shows that in higher rank all irreducible lattices arise this way.

## Explanations

### L1 human

Take a group of real matrices defined by equations, such as the matrices of determinant $1$.
Keep only the matrices whose entries are integers.
The result, such as $\mathrm{SL}_2(\Z)$, is an arithmetic group.
It sits inside the big continuous group like the integer points $\Z^2$ sit inside the plane $\R^2$.
The [[modular group|VEKV1EM77B]] $\mathrm{SL}_2(\Z)$, which acts on the upper half plane, is the most famous example.