In the second half of the talk, we show this abstract set-up can be used to give generic negative solutions to von Neumann s problem (which asks whether or not every small group is amenable) and Day s problem (which asks whether or not every amenable group is elementary amenable). We also discuss the analogous problem for groups satisfying laws and relate it to the well-known open question of whether or not every amenable group satisfying a nontrivial law is uniformly amenable. Finally, we discuss the question of when an amenable group can have the same first-order theory as a nonamenable group.

The work presented in this talk is joint with Srivatsav Kunnawalkam Elayavalli.