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 G is a group. A quasihomomorphism on G is a function f: G \to \R (where \R is the field of real numbers) satisfying the condition that there exists a positive real number D such that for all x,y \in G, we have:

|f(xy) - f(x) - f(y)| \le D

Note that D depends on f, but not on the choice of elements of G.

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 G is a homomorphism. For any quasimorphism f, we can consider its homogenization, defined as \mu_f := x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}.

Examples

  • Any set map from a group to \R with a bounded image is a quasimorphism. In particular, any continuous map from a compact topological group to \R is a quasimorphism. Examples include coordinate projections from compact manifolds embedded in \R^n. 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