Homogeneous quasi-homomorphism space

Description

The homogeneous quasi-homomorphisms is how I call the quotient where each element is equivalent to a homogeneous Quasimorphism.

Definition

Let $G$ be a Discrete group and $\mathcal{V}(G)$ be the vector space of Quasimorphism. Let $BDD(G)$ denote the set of quasi-homomorphisms which are bounded functions. The quotient $QH(G):=\mathcal{V}(G)/BDD(G)$ Denotes the outer quasi-homomorphisms.

Properties

Exact sequence

There is an exact sequence $0\to H^1(G,\mathbb{R})\to QH(G)\to H_b^2(G;\mathbb{R})\to H^2(G:\mathbb{R})$ where $H_b^2$ denotes the second bounded cohomology of $G$.

From this we can follow that $QH(G)$ i.e. the outer quasi-homomorphisms are constructed out of the first cohomology and the second bounded homology. Similarly the second bounded cohomology is constructed out of the quasi-homomorphisms and the second cohomology. If I knew what bounded cohomology is, this could be much more useful.

Contravariance (I think, this is how it is called)

If $G\to G^{\prime }$ is an epimorphism (i.e. a map into a strictly smaller group), then the induced maps $QH(G^{\prime })\to QH(G)$ are injective (i.e. a map into a strictly bigger group) From this follows that big groups have many outer quasi-homomorphisms.

Examples

Trivial and finite groups

It is known, that there are infinitely many quasi-homorphisms on trivial or finite groups. Furthermore the dimensionality is given by the order of the group. However if we divide out all bounded quasi-homomorphisms, the resulting space is just $0$. Much closer to what we want!