We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable infinite-dimensional Hilbert space.2000 Mathematics Subject Classification. Primary 03E15, 54H20; Secondary 20L05, 54H05.
This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information.To effect this enrichment, we show that many of these invariants can be naturally regarded as functors to the category of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification.In the present work we focus on the first derived functors of Hom(−, −) and lim(−). The resulting definable Ext(B, F ) for pairs of countable abelian groups B, F and definable lim 1 (A) for towers A of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable Ext(−, Z) is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups Λ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups Λ with isomorphic classical invariants Ext(Λ, Z). To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover; within this framework we prove several rigidity results for non-Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of the p-adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem R(Aut(Λ)Ext(Λ, Z)) of classifying all group extensions of Λ by Z up to base-free isomorphism, when Λ = Z[1/p] d for prime numbers p and d ≥ 1.
It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.
We represent the universal Menger curve as the topological realization |M| of the projective Fraïssé limit M of the class of all finite connected graphs. We show that M satisfies combinatorial analogues of the Mayer-Oversteegen-Tymchatyn homogeneity theorem and the Anderson-Wilson projective universality theorem. Our arguments involve only 0-dimensional topology and constructions on finite graphs. Using the topological realization M → |M|, we transfer some of these properties to the Menger curve: we prove the approximate projective homogeneity theorem, recover Anderson's finite homogeneity theorem, and prove a variant of Anderson-Wilson's theorem. The finite homogeneity theorem is the first instance of an "injective" homogeneity theorem being proved using the projective Fraïssé method. We indicate how our approach to the Menger curve may extend to higher dimensions.2010 Mathematics Subject Classification. 03C30, 54F15.
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