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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.

AtAs mentioned in the question, at density <1/6$<1/6$, Ollivier--WiseOllivier--Wise showed that a random group is the fundamental group of a compact, non-positively curved cube complex. By Agol's theoremAgol's theorem, such groups are virtually special in the sense of Haglund--WiseHaglund--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--PrzytyckiMackay--Przytycki and MonteeMontee 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).

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.

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.

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).

Source Link
HJRW
  • 25k
  • 3
  • 68
  • 144

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.

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.