Properties (T) and (FA) I have been thinking a lot recently about Property (T) and Property (FA) for discrete groups. I understand that the prior implies the latter, but not the other way around, and I have also seen one or two ad-hoc examples that illustrate this failure. I was just wondering if anything else is known. Is there a sense for how "rare" (FA)-but-no-(T)  is? For instance, is there a theorem that provides a sufficient condition for an (FA) group to have (T)? Otherwise, is there a place I can see a large collection of (FA)-but-no-(T) groups?
Thanks for your time.
 A: Here is a point of view which justifies why Property $(FA)$ is a very particular case of Property $(T)$. First, Chatterji-Drutu-Haglund proved that:
Theorem: A discrete group has $(T)$ iff all its isometric actions on metric median spaces have bounded orbits.
A metric space $(X,d)$ is median if, for every triple $x,y,z \in X$, there exists one and only one point $m \in X$ satisfying
$$\left\{ \begin{array}{l} d(x,y)=d(x,m)+d(m,y) \\ d(x,z)=d(x,m)+d(m,z) \\ d(y,z)=d(y,m)+d(m,z) \end{array} \right.$$
If $(X,d)$ is geodesic, it amounts to saying that $m$ is the unique point that belongs to the intersection between three geodesics connecing $x,y,z$.
Therefore, you can "discretise" Property $(T)$ by introducing:
Definition: A group has Property $(FW)$ if all its actions by automorphisms on median graphs have bounded orbits.
As proved independently by Chepoï and Roller, median graphs coincide with one-skeleta of CAT(0) cube complexes, so you can replace "median graphs" with "CAT(0) cube complexes" in the previous definition. As a consequence, you can introduce a hierarchy of properties:
Definition: Given an $n \geq 1$, a group has Property $(FW_n)$ if all its actions by automorphisms on $n$-dimensional CAT(0) cube complexes have bounded orbits.
If you want, you can also introduce $(FW_\infty)$ for finite-dimensional CAT(0) cube complexes or $(FW_\omega)$ for CAT(0) cube complexes without infinite cubes. In each case, there is something interesting to say. But the key point is that one-dimensional CAT(0) cube complexes coincide with simplicial trees, so $(FW_1)$ actually coincides with $(FA)$.
Conclusion: Property $(FA)$ is the one-dimensional discrete version of Property $(T)$.
$$\begin{array}{ccc} (FA) & & (T) \\ \Updownarrow & & \Updownarrow \\ (FW_1) & \Leftarrow \cdots \Leftarrow (FW_n) \Leftarrow \cdots  \Leftarrow  (FW_\infty) \Leftarrow (FW_\omega) \Leftarrow(FW) \Leftarrow & (FMed) \end{array}$$
