---
tag: 81A0AD379V
kind: concept
name: Quasimorphism
sources:
  - wikipedia: en:69732690@r1348219104
    locator: Quasimorphism
    note: adopted
    mention: wikipedia://en/69732690
    sections: [Summary]
same_as:
  wikidata: Q111941246
  wikipedia: Quasimorphism
inventory: [S5CQVGM5DA]
---

## Summary

In group theory, given a group $G$, a **quasimorphism** (or **quasi-morphism**) is a [[function|0EPAT51DXQ]] $f:G\to\mathbb{R}$ which is additive up to bounded error, i.e. there exists a constant $D\geq 0$ such that $|f(gh)-f(g)-f(h)|\leq D$ for all $g, h\in G$. The least positive value of $D$ for which this inequality is satisfied is called the **defect** of $f$, written as $D(f)$. For a group $G$, quasimorphisms form a [[subspace|00YC1D652K]] of the [[function space|0GZB2SN0PH]] $\mathbb{R}^G$.