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Quasimorphism

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

Examples

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

Homogeneous

A quasimorphism is homogeneous if f ( g n ) = n f ( g ) {\displaystyle f(g^{n})=nf(g)} for all g ∈ G , n ∈ Z {\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 f : G → R {\displaystyle f:G\to \mathbb {R} } is a bounded distance away from a unique homogeneous quasimorphism f ¯ : G → R {\displaystyle {\overline {f}}:G\to \mathbb {R} } , given by :

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

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

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

For example, for α ∈ R {\displaystyle \alpha \in \mathbb {R} } , the map Z → Z : n ↦ ⌊ α n ⌋ {\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 Z → Z {\displaystyle \mathbb {Z} \to \mathbb {Z} } by an appropriate equivalence relation, see Construction of the reals numbers from integers (Eudoxus reals).

Notes

  1. Frigerio (2017), p. 12.

References

Further reading

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