Is there a reasonable random model for selecting a finitely presented group $G$ such that with positive probablity (or even with probability almost $1$) some of the following properties hold:


1. $G$ is residually finite.
2. $G$ is subgroup seperable (LERF).
3. The first $l\_2$ Betti number of $G$ is positive.
4. The cost of $G$ is greater than 1.



Is residual finiteness a property of "many" finitely presented groups?
A random group at density less than 1/6 is known to be the fundamental group of a compact, non-positively curved cube complex, by Ollivier--Wise. By Agol's theorem, all such hyperbolic groups are virtually special, and hence residually finite, QCERF, etc.


