Caltech/UCLA Logic Seminar

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. Extending results of Solecki, Ding and Gao showed that if G is a tame non-archimedean abelian group, then in fact all actions of G are potentially $\Pi^0_6$. That is, they are Borel reducible to a $\Pi^0_6$ orbit equivalence relation. They noted that all previously known examples of such actions were in fact potentially $\Pi^0_3$, and conjectured that their upper bound could be improved to $\Pi^0_3$. We refute this by finding an action of a tame non-archimedean abelian group which is not potentially $\Pi^0_5$. This is joint work with Shaun Allison.