# Difference between revisions of "Quasimorphism"

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− | * The collection of all quasimorphisms on a group <math>G</math> is avector space, with the vector space structure being pointwise addition and scalar multiplication of functions. This vector space is denoted <math>\hat{Q}(G)</math>. The subspace of homogeneous quasimorphisms is denoted <math>Q(G)</math>. The quotient space <math>\hat{Q}(G)/\operatorname | + | * The collection of all quasimorphisms on a group <math>G</math> is avector space, with the vector space structure being pointwise addition and scalar multiplication of functions. This vector space is denoted <math>\hat{Q}(G)</math>. The subspace of homogeneous quasimorphisms is denoted <math>Q(G)</math>. The quotient space <math>\hat{Q}(G)/\operatorname{Hom}(G,\R)</math> is a normed vector space with the defect being the norm. |

==Examples== | ==Examples== |

## Revision as of 00:13, 5 February 2014

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 .

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.

## 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