---
tag: VNJWSCVP6V
kind: concept
name: Fuchsian group
msc: [20H10, 30F35, "22E40"]
specializes: [5M7D1BSHVZ]
sources:
  - url: https://en.wikipedia.org/wiki/Fuchsian_group
same_as:
  wikidata: Q205124
  wikipedia: Fuchsian group
---

## Summary

A Fuchsian group is a discrete group of orientation-preserving isometries of the hyperbolic plane, that is, a discrete subgroup of $\mathrm{PSL}_2(\R)$.
It acts properly discontinuously on the [[upper half-plane|X6XGGH5FPW]], and the quotient is a Riemann surface, possibly with cone points and cusps.
By the uniformization theorem, every compact Riemann surface of genus at least $2$ arises as such a quotient.
The [[modular group|VEKV1EM77B]] and its congruence subgroups are the Fuchsian groups behind modular forms.

## Explanations

### L1 human

The hyperbolic plane can be drawn as the upper half of the complex plane, where lines are half-circles and the scale shrinks near the bottom edge.
A Fuchsian group is a set of rigid motions of this plane that is closed under composition and has no motions arbitrarily close to doing nothing.
Its images of a single tile then cover the plane without crowding, like the famous tilings in Escher's "Circle Limit" pictures.
Gluing the edges of one tile gives a surface, and in this way every closed surface with two or more handles gets a hyperbolic shape.