---
tag: YM8W39B3FX
kind: concept
name: Arithmetic Fuchsian group
aliases: [arithmetic-hyperbolic-surface]
sources:
  - wikipedia: en:50882022@r1374195236
    locator: Arithmetic Fuchsian group
    note: adopted
    mention: wikipedia://en/50882022
    sections: [Summary]
same_as:
  wikidata: Q24894136
  wikipedia: Arithmetic Fuchsian group
inventory: [XGA5TWYHT8]
---

## Summary

**Arithmetic Fuchsian groups** are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group $\mathrm{PSL}_2(\mathbb{Z} )$. They, and the hyperbolic surface associated to their action on the hyperbolic plane, often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.