# Quasimorphism

From Groupprops

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

## Definition

Suppose is a group. A **quasihomomorphism** on is a function (where is the field of real numbers) satisfying the condition that there exists a positive real number such that for all , we have:

Note that depends on , but not on the choice of elements of .

### Homogenization

Ahomogeneous quasimorphism is a quasiomorphism that is also a 1-homomorphism of groups, i.e., its restriction to any cyclic subgroup of is a homomorphism. For any quasimorphism , we can consider its homogenization, defined as .

## Examples

- Any set map from a group to with a bounded image is a quasimorphism. In particular, any continuous map from a compact topological group to is a quasimorphism. Examples include coordinate projections from compact manifolds embedded in . 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.