# Quasimorphism

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

## Contents

## 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 .

The smallest positive real number that works is called the *defect* of the quasimorphism . A quasimorphism of defect 0 is the same as a homomorphism to .

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 is a homomorphism. For any quasimorphism , we can consider its homogenization, defined as .

## Facts

- The collection of all quasimorphisms on a group is avector space, with the vector space structure being pointwise addition and scalar multiplication of functions. This vector space is denoted . The subspace of homogeneous quasimorphisms is denoted . The quotient space is a normed vector space with the defect being the norm.

### Definition in terms of the language of cohomology

Suppose is a group. Consider the cochain complex for the trivial group action of on . In particular:

- is the additive group of all functions from to . In this case, has the structure of a -vector space.
- is the additive group of all functions from to , and also has the structure of a -vector space.

Consider the coboundary map of the cochain complex:

given as:

The standard terminology is as follows:

- The kernel of is the group of 1-cocycles for the trivial group action of on , which coincides with the group . Moreover, the group of 1-coboundaries is trivial, so this also coincides with the first cohomology group . See first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms.
- The image of is the group of 2-coboundaries for the trivial group action of on , and is denoted . This, too, is a -vector space.
- The vector space that we are interested in is the inverse image under of the vector subspace of comprising the 2-coboundaries that are
*bounded*maps to . - By the first isomorphism theorem he quotient space is isomorphic to . Moreover, the defect norm on corresponds with the -norm on this vector space.

## 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.

## External links

- AMS Notices "What Is" page
- Faces of the scl norm ball, a blog post on the
*Geometry and the Imagination*blog