On the Classification Problem for Rank 2 Torsion-Free Abelian Groups

We study here some foundational aspects of the classification problem for torsion-free abelian groups of finite rank. These are, up to isomorphism, the subgroups of the additive groups (Q^n, +), for some n = 1, 2, 3,.... The torsion-free abelian groups of rank ≤ n are the subgroups of (Q^n, +)

Turbulence, amalgamation and generic automorphisms of homogeneous structures

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges-Hodkinson-Lascar-Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many $\omega$-stable, $\aleph_0$-categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group $S_\infty$ has a unique non-trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA)

On a notion of smallness for subsets of the Baire space

Let us call a set A ⊆ ω^ω of functions from ω into ω σ-bounded if there is a countable sequence of functions (α_n: n Є ω)⊆ ω^ω such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of ω^ω. We show that most of the usual definability results about the structure of countable subsets of ω^ω have corresponding versions which hold about σ-bounded subsets of ω^ω. For example, we show that every Σ_(2n+1^1 σ-bounded subset of ω^ω has a Δ_(2n+1)^1 "bound" {α_m: m Є ω} and also that for any n ≥ 0 there are largest σ-bounded Π_(2n+1)^1 and Σ_(2n+2)^1 sets. We need here the axiom of projective determinacy if n ≥ 1. In order to study the notion of σ-boundedness a simple game is devised which plays here a role similar to that of the standard ^*-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the ^*- and ^(**)-(or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of ω^ω whose special cases include countability, being of the first category and σ-boundedness and for which one can generalize all the main results of the present paper

Determinacy with Complicated Strategies

For any class of functions F from R into R, AD(F) is the assertion that in every two person game on integers one of the two players has a winning strategy in the class F. It is shown, in ZF + DC + V = L(R), that for any F of cardinality < 2^(N0)(i.e. any F which is a surjective image of R) AD(F) implies AD (the Axiom of Determinacy)

Minimal Upper Bounds for Sequences of Δ^1_(2n)-Degrees

It is proved here, assuming Projective Determinacy, that every ascending sequence of Δ^1_(2n)-degrees has a minimal strict upper bound but no least strict upper bound. This generalizes a result of Friedman for n = 1

Forcing with Δ perfect trees and minimal Δ-degrees

This paper is a sequel to [3] and it contains, among other things, proofs of the results announced in the last section of that paper. In §1, we use the general method of [3] together with reflection arguments to study the properties of forcing with Δ perfect trees, for certain Spector pointclasses Γ, obtaining as a main result the existence of a continuum of minimal Δ-degrees for such Γ's, under determinacy hypotheses. In particular, using PD, we prove the existence of continuum many minimal Δ^(1)_(2n+1)-degrees, for all n.^(2) Following an idea of Leo Harrington, we extend these results in §2 to show the existence of minimal strict upper bounds for sequences of Δ-degrees which are not too far apart. As a corollary, it is computed that the length of the natural hierarchy of Δ^(1)_(2n+1)-degrees is equal to ω when n ≥ 1. (By results of Sacks and Richter the length of the natural hierarchy of Δ^(1)_(1)-degrees is known to be equal to the first recursively inaccessible ordinal.

Actions of Polish Groups and Classification Problems

We will discuss in this paper some aspects of a general program whose goal is the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This work is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects up to some notion of equivalence by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. This theory, which has been growing rapidly over the last few years, is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of the broad scope of this theory, there are natural interactions of it with other areas of mathematics, such as the theory of topological groups, topological dynamics, ergodic theory and its relationships with the theory of operator algebras, model theory, and recursion theory

Weak containment in the space of actions of a free group

It is shown that the translation action of the free group with n generators on its profinite completion is the maximum, in the sense of weak containment, measure preserving action of this group. Using also a result of Abért–Nikolov this is used to give a new proof of Gaboriau’s theorem that the cost of this group is equal to n. A similar maximality result is proved for generalized shift actions. Finally a study is initiated of the class of residually finite, countable groups for which the finite actions are dense in the space of measure preserving actions

On the concept of π^1_1-completeness

It is shown that two natural notions of completeness for co-analytic sets in Polish spaces, one in terms of continuous reductions and the other in terms of Borel reductions, coincide. The proof uses methods of effective descriptive set theory