Quasimorphism

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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.

Definition in terms of the language of cohomology

Suppose G is a group. Consider the cochain complex C^*(G;\R) for the trivial group action of G on \R. In particular:

  • C^1(G;\R) is the additive group of all functions from G to \R. In this case, C^1(G;\R) has the structure of a \R-vector space.
  • C^2(G;\R) is the additive group of all functions from G \times G to \R, and also has the structure of a \R-vector space.

Consider the coboundary map of the cochain complex:

d_1: C^1(G;\R) \to C^2(G;R)

given as:

d_1(f) := (x,y) \mapsto f(x) + f(y) - f(xy)

The standard terminology is as follows:

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.

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