Generic properties in spaces of enumerated groups

We introduce and study Polish topologies on various spaces of countable enumerated groups. Our study is focused on an abstract class of countable groups which are `locally universal' for these spaces, whose existence and co-meagerness is a consequence of the Baire-category theorem. Hence, by studying properties of these groups, we obtain interesting 'genericity' results such as the following: (1) The generic small group (small meaning the group does not admit nonabelian free subgroups) is nonamenable. (2) The generic amenable group is not elementary amenable. (The above two collectively obtain a `generic negative solution' to the von Neumann-Day problem) (3) The generic amenable group is elementarily equivalent to continuum many nonisomorphic countable nonamenable groups. (4) The generic amenable group cannot have the same first order theory as a group with Property (T). We also provide a connection between genericity in these spaces and model theoretic forcing. We document several open questions in connection with these considerations.