Quasimorphism
In group theory, given a group [math]\displaystyle{ G }[/math], a quasimorphism (or quasi-morphism) is a function [math]\displaystyle{ f:G\to\mathbb{R} }[/math] which is additive up to bounded error, i.e. there exists a constant [math]\displaystyle{ D\geq 0 }[/math] such that [math]\displaystyle{ |f(gh)-f(g)-f(h)|\leq D }[/math] for all [math]\displaystyle{ g, h\in G }[/math]. The least positive value of [math]\displaystyle{ D }[/math] for which this inequality is satisfied is called the defect of [math]\displaystyle{ f }[/math], written as [math]\displaystyle{ D(f) }[/math]. For a group [math]\displaystyle{ G }[/math], quasimorphisms form a subspace of the function space [math]\displaystyle{ \mathbb{R}^G }[/math].
Examples
- Group homomorphisms and bounded functions from [math]\displaystyle{ G }[/math] to [math]\displaystyle{ \mathbb{R} }[/math] 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 [math]\displaystyle{ G=F_S }[/math] be a free group over a set [math]\displaystyle{ S }[/math]. For a reduced word [math]\displaystyle{ w }[/math] in [math]\displaystyle{ S }[/math], we first define the big counting function [math]\displaystyle{ C_w:F_S\to \mathbb{Z}_{\geq 0} }[/math], which returns for [math]\displaystyle{ g\in G }[/math] the number of copies of [math]\displaystyle{ w }[/math] in the reduced representative of [math]\displaystyle{ g }[/math]. Similarly, we define the little counting function [math]\displaystyle{ c_w:F_S\to\mathbb{Z}_{\geq 0} }[/math], returning the maximum number of non-overlapping copies in the reduced representative of [math]\displaystyle{ g }[/math]. For example, [math]\displaystyle{ C_{aa}(aaaa)=3 }[/math] and [math]\displaystyle{ c_{aa}(aaaa)=2 }[/math]. Then, a big counting quasimorphism (resp. little counting quasimorphism) is a function of the form [math]\displaystyle{ H_w(g)=C_w(g)-C_{w^{-1}}(g) }[/math] (resp. [math]\displaystyle{ h_w(g)=c_w(g)-c_{w^{-1}}(g)) }[/math].
- The rotation number [math]\displaystyle{ \text{rot}:\text{Homeo}^+(S^1)\to\mathbb{R} }[/math] is a quasimorphism, where [math]\displaystyle{ \text{Homeo}^+(S^1) }[/math] denotes the orientation-preserving homeomorphisms of the circle.
Homogeneous
A quasimorphism is homogeneous if [math]\displaystyle{ f(g^n)=nf(g) }[/math] for all [math]\displaystyle{ g\in G, n\in \mathbb{Z} }[/math]. It turns out the study of quasimorphisms can be reduced to the study of homogeneous quasimorphisms, as every quasimorphism [math]\displaystyle{ f:G\to\mathbb{R} }[/math] is a bounded distance away from a unique homogeneous quasimorphism [math]\displaystyle{ \overline{f}:G\to\mathbb{R} }[/math], given by :
- [math]\displaystyle{ \overline{f}(g)=\lim_{n\to\infty}\frac{f(g^n)}{n} }[/math].
A homogeneous quasimorphism [math]\displaystyle{ f:G\to\mathbb{R} }[/math] has the following properties:
- It is constant on conjugacy classes, i.e. [math]\displaystyle{ f(g^{-1}hg)=f(h) }[/math] for all [math]\displaystyle{ g, h\in G }[/math],
- If [math]\displaystyle{ G }[/math] is abelian, then [math]\displaystyle{ f }[/math] 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 [math]\displaystyle{ f:G\to\mathbb{Z} }[/math]. In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limit [math]\displaystyle{ \lim_{n\to\infty}f(g^n)/n }[/math] does not exist in [math]\displaystyle{ \mathbb{Z} }[/math] in general.
For example, for [math]\displaystyle{ \alpha\in\mathbb{R} }[/math], the map [math]\displaystyle{ \mathbb{Z}\to\mathbb{Z}:n\mapsto\lfloor\alpha n\rfloor }[/math] is a quasimorphism. There is a construction of the real numbers as a quotient of quasimorphisms [math]\displaystyle{ \mathbb{Z}\to\mathbb{Z} }[/math] by an appropriate equivalence relation, see Construction of the reals numbers from integers (Eudoxus reals).
Notes
- ↑ Frigerio (2017), p. 12.
References
- Calegari, Danny (2009), scl, MSJ Memoirs, 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, 227, American Mathematical Society, Providence, RI, pp. 12–15, doi:10.1090/surv/227, ISBN 978-1-4704-4146-3
Further reading
- What is a Quasi-morphism? by D. Kotschick
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