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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The smallest positive real number <math>D</math> that works is called the ''defect'' of the quasimorphism <math>f</math>. A quasimorphism of defect 0 is the same as a [[homomorphism of groups|homomorphism]] to <math>\R</math>.
  
 
Other names for this concept are '''quasihomomorphism''' (not to be confused with a different notion of [[quasihomomorphism of groups]]) and '''pseudocharacter'''.
 
Other names for this concept are '''quasihomomorphism''' (not to be confused with a different notion of [[quasihomomorphism of groups]]) and '''pseudocharacter'''.
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A [[homogeneous quasimorphism]] is a quasiomorphism that is also a [[1-homomorphism of groups]], i.e., its restriction to any cyclic subgroup of <math>G</math> is a homomorphism. For any quasimorphism <math>f</math>, we can consider its [[homogenization of a quasimorphism|homogenization]], defined as <math>\mu_f := x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}</math>.
 
A [[homogeneous quasimorphism]] is a quasiomorphism that is also a [[1-homomorphism of groups]], i.e., its restriction to any cyclic subgroup of <math>G</math> is a homomorphism. For any quasimorphism <math>f</math>, we can consider its [[homogenization of a quasimorphism|homogenization]], defined as <math>\mu_f := x \mapsto \lim_{n \to \infty} \frac{f(x^n)}{n}</math>.
  
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==Facts==
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* The collection of all quasimorphisms on a group is a
 
==Examples==
 
==Examples==
  
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* [http://www.ams.org/notices/200402/what-is.pdf AMS Notices "What Is" page]
 
* [http://www.ams.org/notices/200402/what-is.pdf AMS Notices "What Is" page]
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* [https://lamington.wordpress.com/2009/08/04/faces-of-the-scl-norm-ball/ Faces of the scl norm ball], a blog post on the ''Geometry and the Imagination'' blog

Revision as of 00:00, 5 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.

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

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}.

Facts

  • The collection of all quasimorphisms on a group is a

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