Linear algebraic groups and countable Borel equivalence relations

If is an equivalence relation on a standard Borel space , then we say that is Borel reducible to if there is a Borel function such that . An equivalence relation on a standard Borel space is Borel if its graph is a Borel subset of . It is countable if each of its equivalence classes is countable. We investigate the complexity of Borel reducibility of countable Borel equivalence relations on standard Borel spaces. We show that it is at least as complex as the relation of inclusion on the collection of Borel subsets of the real line. We also show that Borel reducibility is -complete. The proofs make use of the ergodic theory of linear algebraic groups, and more particularly the superrigidity theory of R. Zimmer.

     

Superrigidity and countable Borel equivalence relations

We formulate a Borel version of a corollary of Furman's superrigidity theorem for orbit equivalence and present a number of applications to the theory of countable Borel equivalence relations. In particular, we prove that the orbit equivalence relations arising from the natural actions of on the projective planes over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.     

Cardinal characteristics and countable Borel equivalence relations

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we introduce a family of properties of countable Borel equivalence relations which correspond to combinatorial cardinal characteristics of the continuum in the same way that Borel boundedness corresponds to the bounding number . We analyze some of the basic behavior of these properties, showing, e.g., that the property corresponding to the splitting number coincides with smoothness. We then settle many of the implication relationships between the properties; these relationships turn out to be closely related to (but not the same as) the Borel Tukey ordering on cardinal characteristics.     

Borel ideals vs. Borel sets of countable relations and trees

For every μ < ω₁, let I_μ be the ideal of all sets S ω^μ whose order type is <ω^μ. If μ = 1, then I₁ is simply the ideal of all finite subsets of ω, which is known to be Σ⁰₂-complete. We show that for every μ < ω₁, I_μ is Σ⁰_2μ-complete. As corollaries to this theorem, we prove that the set WO_ωμ of well orderings Rω × ω of order type <ω^μ is Σ⁰_2μ-complete, the set LP_μ of linear orderings R ω × ω that have a μ-limit point is Σ⁰_2μ+1-complete. Similarly, we determine the exact complexity of the set LT_μ of trees T ^<ωω of Luzin height <μ, the set WR_μ of well-founded partial orderings of height <μ, the set LR_μ of partial orderings of Luzin height <μ, the set WF_μ of well-founded trees T ^<ωω of height <μ(the latter is an old theorem of Luzin). The proofs use the notions of Wadge reducibility and Wadge games. We also present a short proof to a theorem of Luzin and Garland about the relation between the height of ‘the shortest tree’ representing a Borel set and the complexity of the set.     

On asymptotic dimension of countable Abelian groups

We compute the asymptotic dimension of the rationals given with an invariant proper metric. We also show that a countable torsion Abelian group taken with an invariant proper metric has asymptotic dimension zero.     

Abelian group actions and hypersmooth equivalence relations

We show that a Borel action of a standard Borel group which is isomorphic to a sum of a countable abelian group with a countable sum of real lines and circles induces an orbit equivalence relation which is hypersmooth, i.e., Borel reducible to eventual agreement on sequences of reals, and it follows from this result along with the structure theory for locally compact abelian groups that Borel actions of Polish LCA groups induce orbit equivalence relations which are essentially hyperfinite, extending a result of Gao and Jackson and answering a question of Ding and Gao.     

Abelian ideals of Borel subalgebras and affine Weyl groups

This paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a Borel subalgebra. Our main result is a nice uniform formula for the dimension of maximal ideals in these posets. We also obtain results on ad-nilpotent ideals which complete the analysis started in (J. Algebra 225 (2000) 130, 258 (2002) 112).     

On Borel equivalence relations in generalized Baire space

We construct two Borel equivalence relations on the generalized Baire space κ ^κ , κ ^<κ ?=?κ >?ω, with the property that neither of them is Borel reducible to the other. A small modification of the construction shows that the straightforward generalization of the Glimm-Effros dichotomy fails.     

Extensions in the class of countable torsion-free Abelian groups

It is a classical result that for a torsion-free Abelian group A the group $\operatorname {Ext}_{\mathbb {Z}}(A,B)$ is divisible for any Abelian group B. Hence it is of the form for some uniquely determined cardinals r ₀ and r _p . In this paper we clarify when $\operatorname {Ext}_{\mathbb {Z}}(A,B)=0$ and examine the possible values for r ₀ and r _p in case the groups A and B are countable (torsion-free). We also give some methods for constructing torsion-free groups A and B with prescribed cardinals r ₀ and r _p . This is to say that for suitable sequences (r ₀,r _p ∣p∈?) of cardinals we construct torsion-free countable Abelian groups A and B realizing r ₀ and r _p as their invariants of $\operatorname {Ext}_{\mathbb {Z}}(A,B)$ .     

Abelian quotients and orbit sizes of linear groups

Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G ≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least |G/G′| on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result |G/G′| |V| by Aschbacher and Guralnick.     

A confinal family of equivalence relations and Borel ideals generating them

An increasing θ₁-sequence of Borel equivalence relations on a Polish space that is cofinal (in the sense of Borel reducibility) in the family of all Borel equivalence relations is defined as a development of Rosendal’s construction. It is proved that equivalence relations from this sequence are generated by explicitly defined Borel ideals.     

Integrable orbit equivalence rigidity for free groups

It is shown that every accessible group which is integrable orbit equivalent to a free group is virtually free. Moreover, we also show that any integrable orbit-equivalence between finitely generated groups extends to their end compactifications.     

Abelian quotients and orbit sizes of solvable linear groups

Let G be a finite group, and let V be a completely reducible faithful Gmodule. It has been known for a long time that if G is abelian, then G has a regular orbit on V. In this paper we generalize this result as follows. Assuming G to be solvable, we show that G has an orbit of size at least |G/G′| on V. This also strengthens a result of Aschbacher and Guralnick in that situation. Additionally, we prove a similar generalization of the well-known result that if G is nilpotent, then G has an orbit of size at least \(\sqrt {\left| G \right|} \) on V.     

CCZ and EA equivalence between mappings over finite Abelian groups

CCZ- and EA-equivalence, which are originally defined for vectorial Boolean functions, has been extended to mappings between finite abelian groups G and H. We obtain an extension theorem for CCZ-equivalent but not EA-equivalent mappings. Recent results in [2] are improved and generalized.     

Elementary equivalence of automorphism groups of reduced Abelian p-groups

Unbounded reduced Abelian p-groups (p ≥ 3) A ₁ and A ₂ are considered. It is proved that if the automorphism groups Aut A ₁ and Aut A ₂ are elementary equivalent, then the groups A ₁ and A ₂ are equivalent in the second order logic bounded with the final rank of the basic subgroups of A ₁ and A ₂.