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](https://arxiv.org/pdf/2206.14616.pdf) 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](https://mathoverflow.net/a/192098/136187) 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.



Residual finiteness of random groups with property (T)
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](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwj1vejzl4L5AhXHh1wKHXhqDAwQFnoECAMQAQ&url=http%3A%2F%2Fwww.yann-ollivier.org%2Frech%2Fpubls%2Fsixthcubes.pdf&usg=AOvVaw2Gzerup-rJoFvtRF0WWI4y) showed that a random group is the fundamental group of a compact, non-positively curved cube complex. By [Agol's theorem](https://arxiv.org/abs/1204.2810), such groups are virtually special in the sense of [Haglund--Wise](https://link.springer.com/article/10.1007/s00039-007-0629-4), 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](https://arxiv.org/abs/1407.0332) and [Montee](https://arxiv.org/abs/2106.14931) 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](https://arxiv.org/abs/2011.09276) of generalised triangle groups. Some of these have since been [shown](https://arxiv.org/abs/2012.09019) to be residually finite, again by Ashcroft, but since the methods use cubulation these ones certainly don't have (T).


