Difference between revisions of "Quasimorphism"

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(Facts)
(Facts)
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==Facts==
 
==Facts==
  
* 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.
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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{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 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 G is avector space, with the vector space structure being pointwise addition and scalar multiplication of functions. This vector space is denoted \hat{Q}(G). The subspace of homogeneous quasimorphisms is denoted Q(G). The quotient space \hat{Q}(G)/\operatorname{Hom}(G,\R) is a normed vector space with the defect being the norm.

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