# 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 quasimorphism]]s 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. | * 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 quasimorphism]]s 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. | ||

+ | |||

+ | ===Definition in terms of the language of cohomology=== | ||

+ | |||

+ | Suppose <math>G</math> is a [[group]]. Consider the [[cochain complex for a group action|cochain complex]] <math>C^*(G;\R)</math> for the trivial group action of <math>G</math> on <math>\R</math>. In particular: | ||

+ | |||

+ | * <math>C^1(G;\R)</math> is the additive group of all functions from <math>G</math> to <math>\R</math>. In this case, <math>C^1(G;\R)</math> has the structure of a <math>\R</math>-vector space. | ||

+ | * <math>C^2(G;\R)</math> is the additive group of all functions from <math>G \times G</math> to <math>\R</math>, and also has the structure of a <math>\R</math>-vector space. | ||

+ | |||

+ | Consider the coboundary map of the cochain complex: | ||

+ | |||

+ | <math>d_1: C^1(G;\R) \to C^2(G;R)</math> | ||

+ | |||

+ | given as: | ||

+ | |||

+ | <math>d_1(f) := (x,y) \mapsto f(x) + f(y) - f(xy)</math> | ||

+ | |||

+ | The standard terminology is as follows: | ||

+ | |||

+ | * The kernel of <math>d_1</math> is the group of 1-cocycles for the trivial group action of <math>G</math> on <math>\R</math>, which coincides with the group <math>\operatorname{Hom}(G;\R)</math>. Moreover, the group of 1-coboundaries is trivial, so this also coincides with the first cohomology group <math>H^1(G;\R)</math>. See [[first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms]]. | ||

+ | * The image of <math>d_1</math> is the group of [[2-coboundary for a group action|2-coboundaries for the trivial group action]] of <math>G</math> on <math>\R</math>, and is denoted <math>B^2(G;\R)</math>. This, too, is a <math>\R</math>-vector space. | ||

+ | * The vector space <math>\hat{Q}(G)</math> that we are interested in is the inverse image under <math>d_1</math> of the vector subspace <math>B_{bdd}^2(G;\R)</math> of <math>B^2(G;\R)</math> comprising the 2-coboundaries that are ''bounded'' maps to <math>\R</math>. | ||

+ | * By the [[first isomorphism theorem]] he quotient space <math>\hat{Q}(G)/\operatorname{Hom}(G,\R)</math> is isomorphic to <math>B_{bdd}^2(G;\R)</math>. Moreover, the defect norm on <math>\hat{Q}(G)/\operatorname{Hom}(G,\R)</math> corresponds with the <math>L^\infty</math>-norm on this vector space. | ||

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

## Latest revision as of 01:25, 5 February 2014

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