Quasimorphism

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 $f:G\to \mathbb {R}$ (where $\mathbb {R}$ 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:

$|f(xy)-f(x)-f(y)|\leq D$

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 $\mathbb {R}$.

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 ${\hat {Q}}(G)/\operatorname {Hom} (G,\mathbb {R} )$ 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 $C^{*}(G;\mathbb {R} )$ for the trivial group action of $G$ on $\mathbb {R}$. In particular:

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

Consider the coboundary map of the cochain complex:

$d_{1}:C^{1}(G;\mathbb {R} )\to C^{2}(G;R)$

given as:

$d_{1}(f):=(x,y)\mapsto f(x)+f(y)-f(xy)$

The standard terminology is as follows:

• The kernel of $d_{1}$ is the group of 1-cocycles for the trivial group action of $G$ on $\mathbb {R}$, which coincides with the group $\operatorname {Hom} (G;\mathbb {R} )$. Moreover, the group of 1-coboundaries is trivial, so this also coincides with the first cohomology group $H^{1}(G;\mathbb {R} )$. See first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms.
• The image of $d_{1}$ is the group of 2-coboundaries for the trivial group action of $G$ on $\mathbb {R}$, and is denoted $B^{2}(G;\mathbb {R} )$. This, too, is a $\mathbb {R}$-vector space.
• The vector space ${\hat {Q}}(G)$ that we are interested in is the inverse image under $d_{1}$ of the vector subspace $B_{bdd}^{2}(G;\mathbb {R} )$ of $B^{2}(G;\mathbb {R} )$ comprising the 2-coboundaries that are bounded maps to $\mathbb {R}$.
• By the first isomorphism theorem he quotient space ${\hat {Q}}(G)/\operatorname {Hom} (G,\mathbb {R} )$ is isomorphic to $B_{bdd}^{2}(G;\mathbb {R} )$. Moreover, the defect norm on ${\hat {Q}}(G)/\operatorname {Hom} (G,\mathbb {R} )$ corresponds with the $L^{\infty }$-norm on this vector space.

Examples

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

• AMS Notices "What Is" page
• Faces of the scl norm ball, a blog post on the Geometry and the Imagination blog