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.     

Measurable group actions are essentially Borel actions

If a locally compact groupG acts on a Lebesgue probability space (X, λ), it is natural to consider these conditions: (a) each group element preserves the class of λ, and (b) the action function is measurable. The latter is a weakening of the requirement that the action be Borel, providedX has a particular Borel structure as well as the σ-algebra of measurable sets. In this paper, we give an example showing that such an action need not be Borel relative to the given Borel structure, and prove that there is always a conull invariant subset and a new standard Borel structure on that subset for which the action is Borel. This is the meaning of the title. Supported by a University of Colorado Faculty Fellowship.     

Long Borel hierarchies

We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω₂. This implies that ω₁ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω₂. We also show that assuming a large cardinal hypothesis there are models of ZF in which the Borel hierarchy is arbitrarily long. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Hamilton李超代数的Borel子代数

设H(n)(n≥5)为复数域上秩为n的有限维Hamilton单李超代数.通过对正则元的分类,得到H(n)(n=5,6,7)关于典范环面的所有正根系,从而通过确定单根系得到正根系的连接关系,进而得到所有的Borel子代数及其连接关系.证明了H(n)(n≥5)的所有Borel子代数都不是极大可解的子代数.     

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.     

Weak Borel chromatic numbers

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G ‐independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge. We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most ?₁. We observe that some weak version of Todorcevic's Open Coloring Axiom for closed colorings follows from MA. Slightly weaker results hold for F_σ‐graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable F_σ‐graph has a Borel chromatic number of at most ?₁. We refute various reasonable generalizations of these results to hypergraphs (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Principal Borel ideals and Gotzmann ideals

In this paper we characterize all principal Borel ideals with Borel generator up to degree 4 which are Gotzmann. We also classify principal Borel ideals with a Borel generator of degree d which are lexsegment and we describe the shadows of principal Borel ideals. Finally, we discuss the corresponding results for squarefree monomial ideals.Received: 10 May 2002     

Strong conjugation actions induced by flags of exact Borel subalgebras

Let A be a quasi-hereditary algebra with a strong exact Borel subalgebra. It is proved that for any standard semisimple subalgebra T there exists an exact Borel subalgebra B of A such that T is a maximal semisimple subalgebra of B. It is shown that the maximal length of flags of exact Borel subalgebras of A is the difference of the radium and the rank of Grothendic group of A plus 2. The number of conjugation-classes of exact Borel subalgebras is 1 if and only if A is basic; the number is 2 if and only if A is semisimple. For all other cases, this number is 0 or no less than 3. Furthermore, it is shown that all the exact Borel subalgebras are idempotent-conjugate to each other, that is, for any exact Borel subalgebras B and C of A, there exists an idempotent e of A, and an invertible element u of A, such that eBe = u^-1eCeu.     

There are no very meager sets in the model in which both the Borel Conjecture and the dual Borel Conjecture are true

We show that the model for the simultaneous consistency of the Borel Conjecture and the dual Borel Conjecture given in  4 actually satisfies a stronger version of the dual Borel Conjecture: there are no uncountable very meager sets.     

Continuous versus Borel reductions

We present some natural examples of countable Borel equivalence relations E, F with E ≤ _B  F such that there does not exist a continuous reduction from E to F.     

Axial Borel functions

A function from the plane to the plane is axial if it does not change one coordinate. We show that every Borel permutation of the plane is a superposition of 11 Borel axial permutations.     

Borel reducibility as an additive property of domains

We prove that under certain requirements, if E and F are Borel equivalence relations, , is a countable union of Borel sets, and E ↾ X _n , is Borel reducible to F for all n, then E ↾ X is Borel reducible to F. Thus the property of Borel reducibility to F is countably additive as a property of domains. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 189–198.     

Isomorphism and Embedding of Borel Systems on Full Sets

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category, up to isomorphism, there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy ≥t by removing the periodic points and “restricting” to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.     

Some Remarks on Borel Type Ideals

We give new equivalent characterizations for ideals of Borel type. Also, we prove that the regularity of a product of ideals of Borel type is bounded by the sum of the regularities of those ideals.     

The normal form of Borel sets. Part II: Borel sets of infinite rank

For each Borel set of reals of infinite rank A we obtain a “normal form” of A by finding a Borel set Ω such that A and Ω continuously reduce to each other. We do so by defining simple Borel operations which are homomorphic to the ω₁ first Veblen ordinal operations of base ω₁ required to compute the Wadge degree of a Borel set.     

Powers of t -spread principal Borel ideals

We prove that t-spread principal Borel ideals are sequentially Cohen–Macaulay and study their powers. We show that these ideals possess the strong persistence property and compute their limit depth.     

Borel radius and T-Radius of the algebroidal function in the unit disc

Using Ahlfors' theory of covering surface and a type-function, we confirm the existence theorem of a Borel radius and a T-radius for the algebroidal function dealing with multiple values in the unit disc, which briefly extend some results for the algebroidal functions in the complex plane.