Quasimorphism

Description

A quasi-homomorphism seems to me like the dual of a quasi geodesic. It is not a geodesic embedded into a group but the image of a group into $\mathbb{R}$. Alternatively you could maybe think about it as a height map on the group which is additive up to an error term.

Generally speaking “Quasi”-properties are interesting as they give more useful invariants of group. Compare the Quasi-Isometry. The Cayley-Graph is unique up to quasi-isometry.

• Can the the quasi-homomorphism be considered the dual of a quasi-geodesic?

Definition

Let $G$ be a Discrete group. A quasi-homomorphism von $G$ is a function $h:G\to \mathbb{R}$ fulfilling $D(h):={\sup }_{{\gamma }_1,{\gamma }_2\in G}|h({\gamma }_1,{\gamma }_2)-h({\gamma }_1)-h({\gamma }_2|<\infty$ Meaning, if the function is additive up to a constant error term. The constant $D(h)$ is called the defect.

The defect is also sometimes denoted by $\Delta$.

Definition (Vector space)

The space of all quasi-homomorphisms form a Vektorraum, denoted $\mathcal{V}(G)$

Note that the defects of the vector space are usually unbounded.

Properties

Tip

Examples

Trivial and Finite Groups

Let $\{e\}$ be the trivial Group. Then there are infinitely many quasi-homomorphisms. They are $h(e)=x\in \mathbb{R}$ since they fulfill $h(e\cdot e)-(h(e)+h(e))=-x$. Similarly we have infinitely many quasi-homomorphisms for finite groups $G$, furthermore we have $\mathcal{V}(G)\cong {\mathbb{R}}^{|G|}$ For $G=\mathbb{Z}$ we have an infinite dimensional space. As one might imagine, this is not a particularly good tool to study infinite groups, motivating the definition of the Homogeneous quasi-homomorphism space.

Group $G=\mathbb{Z}$

We look at possible examples and counterexamples

• $h(n)=|n|$ is not a quasi-homomorphism because $h(n-n)-(h(n)+h(-n))=2n$, i.e. unbounded.

• $h(n)=n^2$ is not a quasi-homomorphism for the same reason

• $h(n)=n^3$ has $h(n+m)-(h(n)+h(m))$ is unbounded as well, because the function is not additive.

• $h(n)=n/c$ is a homorphism but a pretty boring one.

• $h(n)=\{\begin{array}{ll}10n& 2\mid n\\ n& 2\nmid n\end{array}$ is not a quasi-homomorphism because for $o,e$ odd end even we have $h(o+e)-(h(o)+h(e))=(o+e)-(o+10e)=-9e$. We are not even allowed to move the factors linearly depending on $n$.

  However, we could do something like this $h(n)=\{\begin{array}{ll}n& 2\mid n\\ n-1& 2\nmid n\end{array}$. This is not even injective but $|h(n+m)-(h(n)+h(m))|\le 2$.

  The last examples gives a good way to create new quasi-homorphism. Take an actual homomorphism and move the images by a constant at most $\frac{\Delta (h)}{2}$ and we get a quasi-homomorphism

Counting quasimorphisms give easy easy examples to study.

Counting quasimorphisms

Description

In the special case the counting quasimorphism is a function that counts the occurrences of words in a Free group. It can be generalised for arbitrary hyperbolic groups (and then for groups acting on hyperbolic spaces??)

Definition (Free group)

Let $F$ be the free group of the set $S$. Let $w$ be a reduced word in $S$. The big counting function $C_w$ counts (possibly overlapping) occurrences of $w$: $C_w(g)=\text{Number of copies of w in the reduced representative of g}$ The little counting function $c_w(g)$ counts non-overlapping occurrences of $w$: $c_w(g)=\text{Max number of disjoint copies of w in the reduced representative of g}$ The small and big counting quasimorphisms allow for overlapping or non-overlapping counting of the inverse words: $H_w(g):=C_w(g)-C_{w^{-1}}(g)$ $h_w(g):=c_w(g)-c_{w^{-1}}(g)$ They are a natural example of Quasimorphism

Big counting functions/quasimorphisms are sometimes referred to as Brooks functions/quasimorphisms

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