Are all $\delta$-hyperbolic groups CAT(0)?

Question

In Alessandro Sisto's notes on geometric group theory he mentions that "Many, probably most people in the field" believe that not all $\delta$-hyperbolic groups are CAT(0) groups. Can anything be said as to why this would be the suspicion? It seems completely reasonable to me that some CAT(0) (or dare I say CAT(-1)?) space could be cooked up for any given $\delta$-hyperbolic group.

Are there specific examples of hyperbolic groups for which a reasonable effort has been given to show that they are (or are not) CAT(0) without avail? Or are most hyperbolic group's CAT(0) statuses known?

Answer 1

I think Sapir's group $$ G:=\langle a, b, t; a^t=ab, b^t=ba\rangle $$ is the group you are looking for. Sapir and Drutu claimed (originally citing a result of I. Kapovich but then stating that Minasyan had a proof) that $G$ is hyperbolic, but suggested that it might not be linear (1), (2). They posed this problem in about 2004, and (last time I checked!) this question was still a talking point among geometric group theorists. If I recall correctly, the question of whether it was $\operatorname{CAT}(0)$ was also asked*. In 2013, Button provided the first proof in the literature that this group is hyperbolic, and although he proves that certain similar groups are $\operatorname{CAT}(0)$ he does not show that Sapir's group is $\operatorname{CAT}(0)$ (3). Note that Sapir's group is residually finite (ascending HNN-extensions of free groups are always residually finite).

Another group to consider is a group of M. Kapovich. He constructed a hyperbolic group which is not linear. The paper, allegedly, is this one. However, after a quick flick through, I am having trouble finding this specific result. I do not know if M. Kapovich's group is $\operatorname{CAT}(0)$ or not (and, to be honest, having glanced through the paper it probably is $\operatorname{CAT}(0)$), but it is worth contemplating.

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*Certainly I remember at a conference in summer 2012 trying to naively prove that $G$ was $\operatorname{CAT}(0)$, to no avail!

Answer 2

Quoth Gromov, Asymptotic invariants of infinite groups, 1993, p193:

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"Does every word hyperbolic group admit a discrete cocompact action on some metric geodesic space $X$ with $K(X)<\varepsilon<0$?

It seems that we do not lose or gain much by allowing no locally compact spaces (where cocompact should be replaced by cobounded).

It took me about ten years finally to accept the failure in resolving these question and admit "hyperbolicity" as a permanent definition replacing "coarse hyperbolicity" whose life span I thought would be measured by the time needed to prove "hyperbolicity"="$K<0$".

The first case where the $K<0$ question remains open concerns small cancelation groups. (...)"

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(Note that the Wise and Agol developments made substantial progress on the understanding of small cancelation groups, at least in the $C'(1/6)$ case; however in spite of this I'm not sure they're known yet to be CAT(-1) or even virtually CAT(-1); I'm not even sure about CAT(0) but it's possibly known: I'm not sure whether Wise always gets a cocompact action.)

Answer 3

This does not answer the question, as it is still open, but there are some interesting related things to bring up.

First a group being $\delta$-hyperbolic is a coarse condition but being CAT(0) means that the group actually geometrically acts on a CAT(0) space―a space that is simply connected and has a metric non-positive curvature condition. If you take some hyperbolic group the question is: can I find a CAT(0) space the group acts on or how do I show there is no CAT(0) space the group acts on? There doesn't seem to be a nice universal construction, say coming from modifying Cayley graphs, which produces a CAT(0) space to act on and saying that a group doesn't act on a space, without "obvious" obstructions, is difficult.

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Quasi-isometric to CAT(0) but not CAT(0)

Here is something that is known which should make you skeptical of an affirmative answer to the question: there are groups which are quasi-isometric to CAT(0) groups but are not CAT(0) themselves! So there are groups with no geometric action on a CAT(0) space even though it is quasi-isometric to a CAT(0) space. One class of examples come fundamental groups of graph manifolds. Leeb proved in 3-manifolds with(out) metrics of nonpositive curvature that there are graph manifolds without CAT(0) metrics. Then an arguement that this group has no geometric action on a CAT(0) space (I don't remember the paper but it is by Kapovich and Leeb). It is also known that all graph manifold groups are quasi-isometric to a graph manifold group which is CAT(0). In fact, it is known that all graph manifold groups are quasi-isometric (from graph manifolds without boundary).

Most of the obstructions along these lines essentially come from understanding high rank flats and using the flat torus theorem and other related theorems. These sort of arguments do not apply to hyperbolic groups since they do not have high rank flats/high rank abelian subgroups.

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Random groups, depending on model, are hyperbolic and CAT(0).

There is no single definition of random finitely presented group but in most models that are studied random finitely presented groups are hyperbolic. In the paper Cubulating random groups at density 1/6, by Ollivier and Wise, they show that random groups (using density model with parameter 1/6) will also be CAT(0), in fact CAT(0) cubable. I am guessing there are other results along similar lines.