Quasimorphism

In group theory, given a group

, a quasimorphism (or quasi-morphism) is a function

which is additive up to bounded error, i.e. there exists a constant

for which this inequality is satisfied is called the defect of

, written as

For a group

, quasimorphisms form a subspace of the function space

A quasimorphism is homogeneous if

It turns out the study of quasimorphisms can be reduced to the study of homogeneous quasimorphisms, as every quasimorphism

is a bounded distance away from a unique homogeneous quasimorphism

, given by : A homogeneous quasimorphism

has the following properties: One can also define quasimorphisms similarly in the case of a function

In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limit

α ∈

: n ↦ ⌊ α n ⌋

There is a construction of the real numbers as a quotient of quasimorphisms

by an appropriate equivalence relation, see Construction of the reals numbers from integers (Eudoxus reals).