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.)