Difference between revisions of "Quasimorphism"

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==Definition==
 
==Definition==

Revision as of 23:42, 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.

Homogenization

Ahomogeneous 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.