---
tag: 3A1D37TVF6
kind: concept
name: Small cancellation theory
msc: ["20F06", "20F05"]
sources:
  - url: https://en.wikipedia.org/wiki/Small_cancellation_theory
same_as:
  wikidata: Q7543015
  wikipedia: Small cancellation theory
---

## Summary

Small cancellation theory studies groups given by [[presentations|0JGEQXQ4AC]] whose relators have only short common pieces.
Under such conditions, such as $C'(1/6)$, a nontrivial word equal to the identity must contain a long part of some relator.
This gives Dehn's algorithm for the [[word problem|PBGTTAVMMT]], and finitely presented $C'(1/6)$ groups are [[hyperbolic|JWJ8YPR7EB]].
The theory is also a tool to build groups with unusual properties, as in the construction of [[Tarski monster groups|TE19BGFFF1]].

## Explanations

### L1 human

A presentation lists generators and relators, words that are declared equal to $1$.
If two relators share a long piece, multiplying them can cancel a lot and produce surprising consequences.
Small cancellation conditions forbid this: any piece shared by two relators is a small part of each.
Then the relators behave almost independently, and one can tell when a word equals $1$ by a simple greedy rule.
For example, the fundamental group of a closed orientable surface of genus $2$, $\langle a, b, c, d \mid [a, b][c, d] \rangle$, satisfies the condition $C'(1/6)$.