Journal of Mathematical Logic

We prove Borel superrigidity results for suitably chosen actions of groups of the form SL₂(ℤ[1/p₁, … , 1/p_t]), where {p₁, …, p_t} is a finite nonempty set of primes, and present a number of applications to the theory of countable Borel equivalence relations. In particular, for each prime q, we prove that the orbit equivalence relations arising from the natural actions of SL₂(ℤ[1/q]) on the projective lines ℚ_p ∪ {∞}, p ≠ q, over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

Assuming AD + DC(ℝ), we characterize the self-dual boldface pointclasses which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank , and those of Wadge rank when ξ is limit.

We prove determinacy for open length ω₁ games. Going further we introduce, and prove determinacy for, a stronger class of games of length ω₁, with payoff conditions involving the entire run, the club filter on ω₁, and a sequence of ω₁ disjoint stationary subsets of ω₁. The determinacy proofs use an iterable model with a class of indiscernible Woodin cardinals, and we show that the games precisely capture the theory of the minimal model for this assumption.