# Difference between revisions of "Quasimorphism"

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Note that <math>D</math> depends on <math>f</math>, but not on the choice of elements of <math>G</math>. | Note that <math>D</math> depends on <math>f</math>, but not on the choice of elements of <math>G</math>. | ||

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

===Homogenization=== | ===Homogenization=== | ||

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

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+ | ==External links== | ||

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+ | * [http://www.ams.org/notices/200402/what-is.pdf AMS Notices "What Is" page] |

## Revision as of 23:54, 4 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 .

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 .

## 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.