# Quasimorphism

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with quasihomomorphism of groups

## Definition

Suppose $G$ is a group. A quasihomomorphism on $G$ is a function $f: G \to \R$ (where $\R$ is the field of real numbers) satisfying the condition that there exists a positive real number $D$ such that for all $x,y \in G$, we have:

$|f(xy) - f(x) - f(y)| \le D$

Note that $D$ depends on $f$, but not on the choice of elements of $G$.

The smallest positive real number $D$ that works is called the defect of the quasimorphism $f$. A quasimorphism of defect 0 is the same as a homomorphism to $\R$.

Other names for this concept are quasihomomorphism (not to be confused with a different notion of quasihomomorphism of groups) and pseudocharacter.

### Homogenization

A homogeneous quasimorphism is a quasiomorphism that is also a 1-homomorphism of groups, i.e., its restriction to any cyclic subgroup of $G$ is a homomorphism. For any quasimorphism $f$, we can consider its homogenization, defined as $\mu_f := x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}$.

## Facts

• The collection of all quasimorphisms on a group is a

## Examples

• Any set map from a group to $\R$ with a bounded image is a quasimorphism. In particular, any continuous map from a compact topological group to $\R$ is a quasimorphism. Examples include coordinate projections from compact manifolds embedded in $\R^n$. Note that the homogenization of any such quasimorphism is the zero quasimorphism, so such quasimorphisms are not interesting up to homogenization.
• The rotation number quasimorphism is a homogeneous quasimorphism.