An Invitation to Descriptive Group Theory

Abstract: I will present joint work with I. Goldbring and Y. Lodha where
we study the space of enumerated groups which is a topological Polish
space consisting of enumerations of countable groups. We also study Polish
subspaces of this space consisting of groups with desired properties such
as amenability, orderability etc. The fundamental problem of proving or
disproving existence of co-meager isomorphism classes in these spaces is
surprisingly mysterious. We settle this problem for a family of subspaces
including the space of all countable enumerated groups and the space of
all left orderable enumerated groups. Our work crucially uses tools from
descriptive set theory, combinatorial group theory and model theoretic
forcing, especially ideas of Hodges.