Séminaire Logique mathématique ICJ
# André Nies "Borel classes of closed subgroups of Sym(N)"

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Description

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.