Weak proper discontinuity

Description

This generalises the Properly discontinuous action. If a group acts WPD on a space, then it has a non-trivial Coarse quasi-homomorphism space.

The general idea is something about having two axis of hyperbolic elements which are translates of one another and stay mostly parallel to one-another where one is oriented in the opposite way. This property should be juxtaposted with what happens with axes in free groups (imagine a free Fuchsian group). Here, the translate of an axis is another axis, which points in the same direction and strongly curves away from the first.

Definition

Let $G$ be a group acting simplicially on a $\delta$-hyperbolic complex $X$. The action of an element $g\in G$ is called weakly properly discontinuous if for every $x\in X$ and every $C>0$ there is a constant $N>0$ surch that the set of elements $f\in G$ for which $d_X(x,fx)\le C\text{ and }{d}_X(g^{N}x,f{g}^{N}x)\le C$ is finite.

This is kind of tricky to understand. We try to construct an example which is not WPD. For that we need infinitely many $f$ which fulfil both equations. The first equation tells us, that we should choose a group with the following properties. There is an $x\in X$ and a ball around $x$ with a radius $C$. And there are infinitely many $fx$ which lie in the ball (possibly some finite fixed points as well). Now we apply the action of $g$ on the right of all points $fx$. This has the effect of moving them along the axis $l$ of $g$ and all other axes $fl$. Meanwhile we move the ball without changing its radius. If for all $g^N$ the ball has infinitely many elements inside, the action is not weakly properly discontinuous.

Wow, this sound like a pretty strong property.

Properties

Tip

Examples

Example