Department of Mathematics

Any continuous action of a locally compact Polish group induces an essentially countable orbit equivalence relation, meaning it is Borel reducible to one induced by a countable group. We show that in the case of certain Lie groups acting by conjugation on their space of discrete subgroups, we can realize this reduction in a very concrete fashion; there is a countable subgroup whose action on discrete subgroups is bireducible to that of the larger group. This construction is reminiscent to that of a pure-injective hull from model theory and has implications for the complexity of certain classification problems in geometric topology.

(This is part of joint work with Jeffrey Bergfalk.)