  We introduce and study Polish topologies on various spaces of countable
enumerated groups, where an enumerated group is simply a group whose underlying
set is the set of natural numbers. Using elementary tools and well known
examples from combinatorial group theory, combined with the Baire category
theorem, we obtain a plethora of results demonstrating that several phenomena
in group theory are generic. In effect, we provide a new topological framework
for the analysis of various well known problems in group theory. We also
provide a connection between genericity in these spaces, the word problem for
finitely generated groups and model-theoretic forcing. Using these connections,
we investigate the natural question: when does a certain space of enumerated
groups contain a comeager isomorphism class? We obtain a sufficient condition
that allows us to answer the question in the negative for the space of all
enumerated groups and the space of left orderable enumerated groups. We
document several open questions in connection with these considerations.
