Quasimorphism

In group theory, given a group ${\displaystyle G}$, a quasimorphism (or quasi-morphism) is a function ${\displaystyle f:G\to \mathbb {R} }$ which is additive up to bounded error, i.e. there exists a constant ${\displaystyle D\geq 0}$ such that ${\displaystyle |f(gh)-f(g)-f(h)|\leq D}$ for all ${\displaystyle g,h\in G}$. The least positive value of ${\displaystyle D}$ for which this inequality is satisfied is called the defect of ${\displaystyle f}$, written as ${\displaystyle D(f)}$. For a group ${\displaystyle G}$, quasimorphisms form a subspace of the function space ${\displaystyle \mathbb {R} ^{G}}$.

Examples

• Group homomorphisms and bounded functions from ${\displaystyle G}$ to ${\displaystyle \mathbb {R} }$ are quasimorphisms. The sum of a group homomorphism and a bounded function is also a quasimorphism, and functions of this form are sometimes referred to as "trivial" quasimorphisms.^[1]
• Let ${\displaystyle G=F_{S}}$ be a free group over a set ${\displaystyle S}$. For a reduced word ${\displaystyle w}$ in ${\displaystyle S}$, we first define the big counting function ${\displaystyle C_{w}:F_{S}\to \mathbb {Z} _{\geq 0}}$, which returns for ${\displaystyle g\in G}$ the number of copies of ${\displaystyle w}$ in the reduced representative of ${\displaystyle g}$. Similarly, we define the little counting function ${\displaystyle c_{w}:F_{S}\to \mathbb {Z} _{\geq 0}}$, returning the maximum number of non-overlapping copies in the reduced representative of ${\displaystyle g}$. For example, ${\displaystyle C_{aa}(aaaa)=3}$ and ${\displaystyle c_{aa}(aaaa)=2}$. Then, a big counting quasimorphism (resp. little counting quasimorphism) is a function of the form ${\displaystyle H_{w}(g)=C_{w}(g)-C_{w^{-1}}(g)}$ (resp. ${\displaystyle h_{w}(g)=c_{w}(g)-c_{w^{-1}}(g))}$.
• The rotation number ${\displaystyle {\text{rot}}:{\text{Homeo}}^{+}(S^{1})\to \mathbb {R} }$ is a quasimorphism, where ${\displaystyle {\text{Homeo}}^{+}(S^{1})}$ denotes the orientation-preserving homeomorphisms of the circle.

Homogeneous

A quasimorphism is homogeneous if ${\displaystyle f(g^{n})=nf(g)}$ for all ${\displaystyle g\in G,n\in \mathbb {Z} }$. It turns out the study of quasimorphisms can be reduced to the study of homogeneous quasimorphisms, as every quasimorphism ${\displaystyle f:G\to \mathbb {R} }$ is a bounded distance away from a unique homogeneous quasimorphism ${\displaystyle {\overline {f}}:G\to \mathbb {R} }$, given by :

${\displaystyle {\overline {f}}(g)=\lim _{n\to \infty }{\frac {f(g^{n})}{n}}}$.

A homogeneous quasimorphism ${\displaystyle f:G\to \mathbb {R} }$ has the following properties:

• It is constant on conjugacy classes, i.e. ${\displaystyle f(g^{-1}hg)=f(h)}$ for all ${\displaystyle g,h\in G}$,
• If ${\displaystyle G}$ is abelian, then ${\displaystyle f}$ is a group homomorphism. The above remark implies that in this case all quasimorphisms are "trivial".

Integer-valued

One can also define quasimorphisms similarly in the case of a function ${\displaystyle f:G\to \mathbb {Z} }$. In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limit ${\displaystyle \lim _{n\to \infty }f(g^{n})/n}$ does not exist in ${\displaystyle \mathbb {Z} }$ in general.

For example, for ${\displaystyle \alpha \in \mathbb {R} }$, the map ${\displaystyle \mathbb {Z} \to \mathbb {Z} :n\mapsto \lfloor \alpha n\rfloor }$ is a quasimorphism. There is a construction of the real numbers as a quotient of quasimorphisms ${\displaystyle \mathbb {Z} \to \mathbb {Z} }$ by an appropriate equivalence relation, see Construction of the reals numbers from integers (Eudoxus reals).