Generalised counting quasimorphism

Description

The generalised counting quasimorphism generalises the Counting quasimorphism to Hyperbolic metric space acted upon by a group.

We will generalise the idea to any group acting simplicially on a $\delta$-hyperbolic complex $X$ (like the Cayley-Graph). Instead of a word we try to count copies of paths $\sigma$ along a geodesic in the complex (with copies being defined as translates $g\sigma$ for $g\in G$). Before we searched for copies of a word (think: paths) that occur between $1$ and $g$. Now we do the same, but since now there are multiple possible ways to get from $1\cdot p$ to $g\cdot p$, we define a function which takes all possible paths into account.

Definition (Generalisation for groups acting on a hyperbolic complex)

Let $\sigma$ be a finite oriented simplicial path in $X$ and let $p\in X$ be a base vertex. For any oriented simplicial path $\gamma$ in $X$, let $|\gamma {|}_{\sigma }$ denote the maximal number of disjoint copies of $\sigma$ contained in $\gamma$.

Given $p,q\in X$, define the small counting function: $c_{\sigma }([p,q])=d(p,q)-{\inf }_{\gamma }(\text{length}(\gamma )-|\gamma {|}_{\sigma })$
Or for a $p\in X,g\in G$ $c_{\sigma }(g)=d(p,gp)-{\inf }_{\gamma }(\text{length}(\gamma )-|\gamma {|}_{\sigma })$
where the infimum goes over all oriented simplicial paths $\gamma$ in $X$ from $p$ to $q$ or from $p$ to $gp$ respectively. Define the small counting quasimorphism $h_{\sigma }$ as $h_{\sigma }(g)=c_{\sigma }(g)-c_{{\sigma }^{-1}}(g)$ Similarly we define a big counting function and quasimorphism.

Definition (realising path)

The infimum $\text{length}(\gamma )-|\gamma {|}_{\sigma }$ is realised for some path $\gamma$. If so, we call it the realising path for $c_{\sigma }$.

I was wondering, why we need an infimum and why go over all curves? I think, this can be seen if we take the Cayley Graph as a space to act on. The Cayley graph is not well defined and shortest curves might be different, depending on the generating set. But if you allow longer curves, then the differences likely vanish. The reason we we take the difference is likely due to the big counting function. Here the shortest paths might not have $n$ occurrences of $\sigma$ when you increase the path length by $1$, then the occurrences jump to $n+1$. The difference however stays the same. This is not a formal argument.

Properties

Here’s the first properties that sheds light on the connection between quasimorphism and quasigeodesics

Realising paths are geodesics

Suppose $\text{length}(\sigma )\ge 2$. Any realising path for $c_{\sigma }$ is a $(k,ϵ)$ quasi-geodesic with $k=\frac{\text{length}(\sigma )}{\text{length}(\sigma )-1},ϵ=\frac{2\cdot \text{length}(\sigma )}{\text{length}(\sigma )-1}$ Those values are greatest when the length is $2$

From the Morse lemma we know that for length greater than $2$, every realising path must lie within $C(\delta )$ of a geodesic. This gives us a condition when the counting function vanishes. Or more interestingly, where we have to search for such quasimorphisms.

Vanishing counting map

Let $X$ be a $\delta$-hyperbolic space. For $p$ and $gp$, if there are no copies of a path $\sigma$ in the $\delta$-neighbourhood of all geodesics then $c_{\sigma (g)}=0$.

Is a quasimorphism

The counting quasimorphism is actually a Quasimorphism as for $\sigma$ with length $\ge 2$ there is a constant $C(\delta )$ such that $D(h_{\sigma })<C$.

Proof: Since the length is greater than $2$ we know that the realising curve will be a $(2,4)$-quasigeodesic. We generalise the theorem to the case of three points. We show that the counting morphism along the realising curves between the three points is bounded. By the Morse-Lemma it the realising curves are close to geodesics and by the hyperbolicity they are close to one another. Specifically, there is a radius $N(\delta )$ around one point of a realising curve covering every other point of the realising curves. Doing a few equation, we can find an estimate $D\le 6+6N$.

• Are the counting morphisms of length $1$ those which are trivial? Compare the free group with the homomorphism defined by $a\mapsto 1$.

Examples

The special case

Counting quasimorphism

Nontrivial example $w=aa$

Let $a\in S$ and $w=aa$. We want to find out if this quasi-morphism is coarsely trivial, meaning it is $0$ in the Coarse quasi-homomorphism space. It is trivial if there is it is the sum of a bounded quasimorphism and a homomorphism. A homomorphism is already determined by all generators. The only sensible option is to send every generator to $\mapsto 0$ and $a\mapsto 1/2$ such that $aa\mapsto 1$.
However, note that in that case the word $(ab{)}^n$ is sent to $n$ under the homomorphism and to $0$ under the quasimorphism. They differ by an unbounded value.

Here’s a class of words which seems to sit between case $1$ and $2$ of the above classification.

Monotone words

A word is monotone if for each $a\in S$ at most one of a $a$ and $a^{-1}$ appears in $w$. Every monotone word sits in class 1 of the above classification, meaning $D(h_w)\le 1$.