---
tag: KHEJ0FVZXT
kind: concept
name: Curve complex
msc: [57K20, 20F65, 30F60]
specializes: [V8GZ0W1ZZ5]
sources:
  - url: https://en.wikipedia.org/wiki/Curve_complex
same_as:
  wikidata: Q28455227
  wikipedia: Curve complex
---

## Summary

The curve complex of a [[surface|VGJJVBZ1MQ]] is a [[simplicial complex|V8GZ0W1ZZ5]].
Its vertices are the isotopy classes of essential simple closed curves, and its simplices are sets of curves that can be made pairwise disjoint.
The [[mapping class group|JV5NYG913V]] acts on it by simplicial automorphisms.
Harvey introduced it, and Masur and Minsky proved that it is hyperbolic in the sense of Gromov.
It is a central tool in the study of mapping class groups and of Teichmüller space.

## Explanations

### L1 human

Draw all the essentially different loops on a surface, one dot for each.
Join two dots by an edge when the two loops can be drawn without crossing.
Fill in a triangle when three loops can be drawn pairwise without crossing, and so on.
The resulting shape is infinite and tangled, but it records how all loops on the surface sit with respect to each other.