---
tag: YBERRJMXT6
kind: concept
name: Schottky group
msc: [30F40, 30F10, "20E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Schottky_group
same_as:
  wikidata: Q7432778
  wikipedia: Schottky group
---

## Summary

A Schottky group is a group of [[Möbius transformations|0SG74WBFA2]] generated by maps that pair off disjoint discs in the [[Riemann sphere|0J78KBPFZB]].
It is a Kleinian group that is a [[free group|0KTW26C80J]] and has only loxodromic elements besides the identity.
The quotient of its domain of discontinuity is a compact [[Riemann surface|5K0N8QMV6Q]].
Every compact Riemann surface arises in this way, which gives a concrete way to build them.

## Explanations

### L1 human

Draw $2g$ disjoint circles on the Riemann sphere and pair them up.
For each pair, choose a Möbius map that sends the outside of the first circle onto the inside of the second.
The group generated by these $g$ maps moves the region outside all the circles around without overlap.
Gluing each circle to its partner turns that region into a surface with $g$ handles.
For $g = 1$ and the map $z \mapsto 2z$, with circles $\abs{z} = 1$ and $\abs{z} = 2$, the glued annulus is a torus.