• 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.

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".

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).

• 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

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