---
tag: CEHHBBD6NT
kind: concept
name: Translation surface
msc: ["37E35", "30F30", "32G15"]
sources:
  - url: https://en.wikipedia.org/wiki/Translation_surface
same_as:
  wikidata: Q2449377
  wikipedia: Translation surface
---

## Summary

A translation surface is a closed surface with a flat metric whose charts change by translations, except at finitely many cone points.
Equivalently, it is a compact [[Riemann surface|5K0N8QMV6Q]] with a nonzero holomorphic 1-form, an [[abelian differential|P014C13K5H]].
It can also be given by finitely many polygons in the plane with sides glued in pairs by translations.
Straight-line flows on translation surfaces generalize linear flows on the torus and are closely tied to [[interval exchange transformations|19JFF3GY9F]] and to billiards in rational polygons.
The space of translation surfaces carries an action of $SL(2, \R)$, whose study links dynamics with [[Teichmüller theory|ZZGC6BZNJ5]].

## Explanations

### L1 human

Take a square of paper and glue opposite sides: you get a torus that is flat everywhere, on which straight lines make sense.
Do the same with a regular octagon, gluing each side to the opposite one, and you get a surface with two holes.
It is flat everywhere except at one point, where all eight corners meet and the angles add up to three full turns instead of one.
Such surfaces, made of polygons glued by translations, are translation surfaces.