Quasimorphism

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

Examples

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

Homogeneous

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

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

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

• It is constant on conjugacy classes, i.e. $f(g^{-1}hg)=f(h)$ for all $g,h\in G$,
• If $G$ is abelian, then $f$ 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 $f:G\to \mathbb {Z}$. In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limit $\lim _{n\to \infty }f(g^{n})/n$ does not exist in $\mathbb {Z}$ in general.

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

References

• Calegari, Danny (2009), scl, MSJ Memoirs, vol. 20, Mathematical Society of Japan, Tokyo, pp. 17–25, doi:10.1142/e018, ISBN 978-4-931469-53-2
• Frigerio, Roberto (2017), Bounded cohomology of discrete groups, Mathematical Surveys and Monographs, vol. 227, American Mathematical Society, Providence, RI, pp. 12–15, arXiv:1610.08339, doi:10.1090/surv/227, ISBN 978-1-4704-4146-3, S2CID 53640921

Further reading

• What is a Quasi-morphism? by D. Kotschick