Borel classes of closed subgroups of Sym(N)
We study various conjugation invariant Borel classes from a logician’s
point of view. The locally Roelcke precompact groups form the largest
class considered. Interesting subclasses include the totally
disconnected locally compact groups, and the oligomorphic groups. We
establish for each class a Borel duality with a class of countable
structures that are based on Roelcke precompact open cosets.
This is used for an upper bound on the Borel complexity of
topological isomorphism relations (with Schlicht and Tent), and for an
algorithmic theory in the t.d.l.c. case. A lower bound on the
complexity of topological isomorphism remains open for the
oligomorphic groups. Paolini and Shelah obtained smoothness under the
additional hypothesis that each open subgroup has the pointwise
stabiliser of a finite set as a subgroup of finite index.