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:
- $G$ is residually finite.
- $G$ is subgroup seperable (LERF).
- The first $l_2$ Betti number of $G$ is positive.
- The cost of $G$ is greater than 1.