Coarse quasi-homomorphism space

Description

We want to study the space of Quasimorphism. Intuitively, you would study $\mathcal{V}(G)$ which is the vector space of quasimorphisms, but this is kind of weird. Quasimorphisms are coarse objects, so it wouldn’t make sense to look at each one of them. It would be like doing cosmology with a microscope. Instead we look at equivalence classes of Quasimorphisms which are the same under a scaling and some noise (similarly to the definition of the Quasi-Isometry). The resulting space is called by me the the Coarse space of quasi-morphisms

• Is the terminology “coarse space” the right way to think about it?
• Why is $H^1(G;\mathbb{R})=HOM(G)$? Is this the first cohomology?

Definition

Let $G$ be a Discrete group and $\mathcal{V}(G)$ the space of Quasimorphism. Let $BDD(G)$ denote the subset of bounded quasi homomorphisms and $H^1(G;\mathbb{R})=HOM(G)$ the subset actual group homomorphisms. The only map sitting in both sets is $g\in G\mapsto 0$. We define the Modulo space of outer quasi-homomorphisms as follows: $\widetilde{QH(G)}:=\mathcal{V}(G)/(BDD(G)+HOM(G))\cong QH(G)/H^1(G;\mathbb{R})$ where $QH(G):=\mathcal{V}(G)/BDD(G)$ are the Homogeneous quasi-homomorphism space.

Motivation

We present a way to create Quasimorphism from an actual Group homomorphism. Let $h:G\to \mathbb{R}$ be an group homomorphism. Now we can move around the images independently $h(g)$ by a constant at most $\frac{\Delta (h)}{2}$. The resulting map $\tilde{h}$ is a quasi-homomorphism with defect $\le \Delta (h)$.
This construction is very easy and doesn’t teach us a lot about the group. We could just as much study the homeomorphism group. The idea is, we will make $\tilde{h}$ unimportant: Consider the map $\tilde{h}-h$. This is a bounded homomorphism, i.e. $\tilde{h}-h\in BDD(G)$. Thereby $\tilde{h}$ sits in the space $BDD(G)+HOM(G)$ and projects to $0$ in $\widetilde{QH(G)}$.

In other words, $0$ in $\widetilde{QH(G)}$ represents all easily constructible quasi-morphisms. Or you could say each element in $\widetilde{QH(G)}$ represents a class where each element can be transformed by adding a homomorphism and a small perturbation.

It represents the interesting classes of quasi-homomorphisms!

Properties

Exact sequence

Due to the exact sequence of the Homogeneous quasi-homomorphism space and since $\widetilde{QH(G)}=QH(G)/H^1(G;\mathbb{R})$ i.e. we have the Kurze Exakte Sequenz $0\to H^1(G;\mathbb{R})\to QH(G)\to \widetilde{QH(G)}\to 0$ we also get the exact sequence $0\to \widetilde{QH(G)}\to H_b^2(G;\mathbb{R})\to H^2(G;\mathbb{R})$ Or $\widetilde{QH(G)}\cong H^2(G;\mathbb{R})/H_b^2(G;\mathbb{R})$

Probably this is just trivial if you know exact sequences.

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 $\widetilde{QH(G^{\prime })}\to \widetilde{QH(G)}$ are injective (i.e. a map into a strictly bigger group) From this follows that big groups have big modulo space.

Examples

In all examples calculated to date $\widetilde{QH(G)}$ is either $0$ or infinite dimensional. Note: 0 means that every Quasimorphism is an actual homomorphism or bounded or there is only the trivial quasi-homomorphism (e.g. $G$ finite).

Calculations on $\widetilde{QH(G)}$

$\widetilde{QH(G)}$ is $0$ when

• $G$ is an Amenable Group
• $G$ is a cocompact irreducible lattice in a semisimple Lie group of real rank $>1$

$\widetilde{QH(G)}$ is infinite dimensional when

• $G$ is a non-abelian free group
• $G$ acts on hyperbolic spaces with some necessary additional conditions

Group acting on Hyperbolic space with $\widetilde{QH(G)}=0$

Take an Irreducible cocompact Lattice in $SL(2,\mathbb{Z})\times SL(2,\mathbb{Z})$ acting discretely on the product ${\mathbb{H}}^2\times {\mathbb{H}}^2$.

• What is a irreducible cocompact lattice?

Mapping class group

Let $G=MCG(S)$ be a Mapping class group. If it is not virtually abelian then $\widetilde{QH(G)}$ is infinite-dimensional.

Groups acting on nonelementary without parallels on hyperbolic graphs

If a group acts nonelementary with two elements $g_1,g_2$ which do not have parallel quasi-axis then the space of coarse quasimorphisms is infinite-dimensional. Special examples of this theorem include:

• $G$ is a Hyperbolic group acting on its Cayley-Graph
• The action of $G$ on $X$ is properly discontinuous. (e.g. Fuchsian group)
• $G$ is a Graph of groups acting on the associated Bass-Serre tree.