---
tag: WVYYQCFS4Q
kind: definition
name: Schottky group
msc: [30F40, 30F10, "20E05"]
defines: YBERRJMXT6
uses: [0SG74WBFA2, 0J78KBPFZB, 0KTW26C80J]
sources:
  - url: https://en.wikipedia.org/wiki/Schottky_group
---

## Definition

Let $g \ge 1$ be an integer.
Let $D_1, D_1', \ldots, D_g, D_g'$ be $2g$ open Jordan discs in the [[Riemann sphere|0J78KBPFZB]] $\hat{\C}$ whose closures are pairwise disjoint.
Let $C_i$ and $C_i'$ be the boundary curves of $D_i$ and $D_i'$.
For each $i$, let $A_i$ be a [[Möbius transformation|0SG74WBFA2]] that maps $C_i$ onto $C_i'$ and maps $\hat{\C} \setminus \overline{D_i}$ onto $D_i'$.
The group generated by $A_1, \ldots, A_g$ is a *Schottky group* of genus $g$.
It is a *classical* Schottky group if the curves $C_i$ and $C_i'$ can be chosen to be circles.

## Remarks

Example: let $\lambda \in \C$ with $\abs{\lambda} > 1$, $A_1(z) = \lambda z$, $D_1 = \set{\abs{z} < 1}$ and $D_1' = \set{\abs{z} > \abs{\lambda}} \cup \set{\infty}$.
Their closures are disjoint, and $A_1$ maps the circle $\abs{z} = 1$ onto $\abs{z} = \abs{\lambda}$ and the region $\abs{z} > 1$ onto $D_1'$.
So $\langle A_1 \rangle$ is a Schottky group of genus $1$; the quotient of $\C \setminus \set{0}$ by it is a torus.

Non-example: the group generated by $z \mapsto z + 1$ is not a Schottky group.
Its generator is parabolic, while every element other than the identity of a Schottky group is loxodromic.

A Schottky group of genus $g$ is a [[free group|0KTW26C80J]] of rank $g$ on $A_1, \ldots, A_g$ and is a Kleinian group.
The common exterior of the $2g$ discs is a fundamental domain for its action on the domain of discontinuity $\Omega$.
The quotient $\Omega / G$ is a compact Riemann surface of genus $g$.