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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.

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    $\begingroup$ You should look at the random groups in the Gromov density model and see if the density for (FA) is not the same as for (T). See this paper. $\endgroup$
    – guest5781
    Commented May 21, 2021 at 1:26
  • $\begingroup$ The wikipedia article has quite a bit of information about groups with property FA but not property T. For example $SL_2({\mathbb Z}[{\sqrt 2}])$ has property FA but not property T. en.wikipedia.org/wiki/Serre%27s_property_FA $\endgroup$
    – Venkataramana
    Commented May 21, 2021 at 3:50
  • $\begingroup$ Many groups acting non-trivially on CAT(0) cube complexes (in particular, they do not have (T)) have (FA). See for instance: mathoverflow.net/questions/308865/… $\endgroup$
    – AGenevois
    Commented May 21, 2021 at 5:09
  • $\begingroup$ Some of the most classical examples that illustrate the difference are the fundamental groups of non-Haken 3-manifolds. These have Property (FA), but never (T). $\endgroup$
    – HJRW
    Commented May 21, 2021 at 9:11
  • $\begingroup$ @guest5781 cites a paper of Dahmani--Guirardel--Przytycki, who proved that random groups in the Gromov density model have property (FA) at all positive densities. By way of contrast, (T) is known to hold at density greater than 1/3 (by a theorem of Zuk--Kotowski--Kotowski), and known not to hold at densities <5/24 (by a theorem of Mackay--Przytycki). So groups random groups at densities <5/24 also provide examples. $\endgroup$
    – HJRW
    Commented May 21, 2021 at 16:24

1 Answer 1

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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 $(T)$.

$$\begin{array}{ccc} (FA) & & (T) \\ \Updownarrow & & \Updownarrow \\ (FW_1) & \Leftarrow \cdots \Leftarrow (FW_n) \Leftarrow \cdots \Leftarrow (FW_\infty) \Leftarrow (FW_\omega) \Leftarrow(FW) \Leftarrow & (FMed) \end{array}$$

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  • $\begingroup$ I've used in this paper the terminology "Property FM" for a property strictly weaker than Property T (every action on a discrete set preserving a mean has a finite orbit). $\endgroup$
    – YCor
    Commented May 21, 2021 at 6:44
  • $\begingroup$ The result mentioned is mostly due to Robertson-Steger for actions of discrete groups (the most difficult part is to pass from an action on a Hilbert to an action on $L^1$-space, which is median — CDH did it for actions of locally compact groups, for which there was a technical difficulty). At that time (around 2005) the notion of "measured wall spaces" was fashionable and the characterization of Property T in these terms was already known (for discrete groups, and explicitly open for locally compact groups). $\endgroup$
    – YCor
    Commented May 21, 2021 at 6:48
  • $\begingroup$ Are there known groups that separate all the properties in this hierarchy? $\endgroup$
    – Aurel
    Commented May 21, 2021 at 7:46
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    $\begingroup$ @Aurel: $SL_2(\mathbf{Z}[\sqrt{2}]$ is FW but not T. The group $(\mathbf{Z}^n)_0\rtimes Sym(n)$ (index zero means one looks at $n$-tuples of sum zero) has FW$_{k(n)-1}$ but not FW$_{k(n)}$ for some $k(n)$, and certainly $k(n)$ tends to infinity (I'm not sure of its value), so certainly this infinite family is "essentially" separated, and with some effort it should be entirely separated using virtually abelian groups. The Grigorchuk group has FW$_\infty$ but not FW, I'm not sure about FW$_\omega$. $\endgroup$
    – YCor
    Commented May 21, 2021 at 8:12
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    $\begingroup$ @Aurel: Finitely generated torsion groups have $(FW_\omega)$ but they may not have $(FW)$, e.g. some infinite free Burnside groups. Thompson's groups $T$ and $V$ have $(FW_\infty)$ but they do not have $(FW_\omega)$. $\endgroup$
    – AGenevois
    Commented May 21, 2021 at 9:07

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