---
tag: 68AFPHR4GN
kind: concept
name: One-relator group
msc: ["20F05", "20F06"]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/One-relator_group
same_as:
  wikidata: Q106901197
  wikipedia: One-relator group
---

## Summary

A one-relator group is a [[group|G5S1M9]] with a [[presentation|0JGEQXQ4AC]] that has a single defining relator.
Examples are $\Z^2$, the fundamental groups of closed surfaces, and the [[Baumslag–Solitar groups|429JJ5ZNBY]].
They are the natural next class after [[free groups|0KTW26C80J]], and they have a rich general theory.
Magnus proved the [[Freiheitssatz|B2TRERFWEH]] for them and showed that their [[word problem|PBGTTAVMMT]] is solvable.

## Explanations

### L1 human

Take a free group, where no rule holds between the generators, and impose exactly one rule.
For example, $\langle a, b \mid a b a^{-1} b^{-1} \rangle$ imposes $ab = ba$, and the result is the group $\Z^2$ of pairs of integers.
One rule is little, so these groups stay close to free groups.
Still they include many important groups, such as the fundamental groups of closed surfaces.