Residual finiteness of random groups with property (T)

Question

A well known result of A. Zuk states that for $\frac{1}{3} < d < \frac{1}{2}$, a random group $\Gamma$ with respect to Gromov's density model with density $d$ has Kazhdan's property (T) with overwhelming probability.

On the other hand, C. J. Ashcroft has recently proved that that at densities below $\frac{1}{4}$, with overwhelming probabilty, $\Gamma$ acts with unbounded orbits on a finite dimensional CAT(0) cube complex, and hence does not have Property (T).

It is also known that below density $\frac{1}{6}$, with overwhelming probability, $\Gamma$ will be residually finite (by deep results of Agol, Ollivier-Wise).

My question is whether there are ranges of densities $d$ where a random group $\Gamma$ sampled according to Gromov's density model at density $d$ is known to be both Kazhdan and residually finite.

Any reference/solution will be greatly appreciated.

Answer 1

This is an open question: there are no densities $1/2>d\geq 1/6$ where a random group is known to be residually finite. Any progress would be a major step forward.

As mentioned in the question, at density $<1/6$, Ollivier--Wise showed that a random group is the fundamental group of a compact, non-positively curved cube complex. By Agol's theorem, such groups are virtually special in the sense of Haglund--Wise, and in particular residually finite.

At density $\geq 1/2$, random groups are a.a.s. finite.

Ashcroft's paper mentioned in the question, like the previous work of Mackay--Przytycki and Montee improving the Ollivier--Wise bound, gives a non-trivial action on a cube complex (which is enough to contradict (T)), but not the proper, cocompact action needed to apply Agol's theorem.

The bottom line is that, unless a hyperbolic group with (T) happens to be linear, we have no tools to prove residual finiteness. Away from the context of random groups, a very concrete class of examples is provided by the recent Caprace--Conder--Kaluba--Witzel census of generalised triangle groups. Some of these have since been shown to be residually finite, again by Ashcroft, but since the methods use cubulation these ones certainly don't have (T).