---
tag: 57TCAKSBQT
kind: definition
name: Fuchsian group
msc: [20H10, 30F35, "22E40"]
defines: VNJWSCVP6V
uses: [5M7D1BSHVZ, Q0GBCB9RA4, X6XGGH5FPW, 0SG74WBFA2, J5X5ED0B2Y]
sources:
  - url: https://en.wikipedia.org/wiki/Fuchsian_group
---

## Definition

Let $\mathrm{PSL}_2(\R) = \mathrm{SL}_2(\R) / \set{\pm I}$, where [[SL(2,R)|Q0GBCB9RA4]] has the topology of a subset of $\R^4$ and $\mathrm{PSL}_2(\R)$ has the quotient topology.
A *Fuchsian group* is a [[discrete subgroup|5M7D1BSHVZ]] of $\mathrm{PSL}_2(\R)$.

## Remarks

The group $\mathrm{PSL}_2(\R)$ acts on the [[upper half-plane|X6XGGH5FPW]] $\mathbb H$ by [[Möbius transformations|0SG74WBFA2]] $z \mapsto (az + b)/(cz + d)$, and this is the group of orientation-preserving isometries of the hyperbolic metric.
A subgroup $\Gamma \subseteq \mathrm{PSL}_2(\R)$ is Fuchsian if and only if its action on $\mathbb H$ is [[properly discontinuous|J5X5ED0B2Y]].

Example: $\mathrm{PSL}_2(\Z)$, the image of the integer matrices of determinant $1$, is Fuchsian.
Two distinct integer matrices differ by at least $1$ in some entry, so a small neighbourhood of $\pm I$ contains no other element of the group.

Non-example: the group generated by $z \mapsto z + 1$ and $z \mapsto z + \sqrt2$ consists of the translations $z \mapsto z + a + b\sqrt2$ with $a, b \in \Z$.
The translation lengths $a + b\sqrt2$ come arbitrarily close to $0$ without being $0$, so the group is not discrete, and it is not Fuchsian.

Some authors call a group Fuchsian when it is a discrete group of Möbius transformations that preserves a disc or half-plane in the Riemann sphere.
Such a group is conjugate to a group as above.
Since $\mathrm{PSL}_2(\R) \subset \mathrm{PSL}_2(\C)$, every Fuchsian group is a Kleinian group.