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 $D\ge 0$ such that $|f(gh)-f(g)-f(h)|\le D$ 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 $C_w:F_S\to {\mathbb{\mathbb{Z}}}_{\ge 0}$, 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 $c_w:F_S\to {\mathbb{\mathbb{Z}}}_{\ge 0}$, 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 $g\in G,n\in \mathbb{\mathbb{Z}}$. 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 $\bar{f}:G\to \mathbb{ℝ}$, given by :

$\bar{f}(g)=\underset{n\to \mathrm{\infty }}{\lim }\frac{f(g^n)}{n}$.

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 $f:G\to \mathbb{\mathbb{Z}}$. In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limit $\underset{n\to \mathrm{\infty }}{\lim }f(g^n)/n$ does not exist in $\mathbb{\mathbb{Z}}$ in general. For example, for $\alpha \in \mathbb{ℝ}$, the map $\mathbb{\mathbb{Z}}\to \mathbb{\mathbb{Z}}:n\mapsto \lfloor \alpha n\rfloor$ is a quasimorphism. There is a construction of the real numbers as a quotient of quasimorphisms $\mathbb{\mathbb{Z}}\to \mathbb{\mathbb{Z}}$ 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