7,156 research outputs found
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 for instance 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
Applications of tree decompositions and accessibility to treeability of Borel graphs
A framework to handle tree decompositions of the components of a Borel graph
in a Borel fashion is introduced, along the lines of Tserunyan's Stallings
Theorem for equivalence relations arXiv:1805.09506. This setting leads to a
notion of accessibility for Borel graphs, together with a treeability
criterion. This criterion is applied to show that, in particular, Borel
equivalence relations associated to Borel graphs with accessible planar
connected components are measure treeable, generalising results of Conley,
Gaboriau, Marks, and Tucker-Drob arXiv:2104.07431 and Timar arXiv:1910.01307.
It is also proven that uniformly locally finite Borel graphs with components of
finite tree-width yield Borel treeable equivalence relations. Our results imply
that p.m.p countable Borel equivalence relations with measured property (T) do
not admit locally finite graphings with planar components a.s.Comment: 39 page
Borel asymptotic dimension and hyperfinite equivalence relations
A long standing open problem in the theory of hyperfinite equivalence
relations asks if the orbit equivalence relation generated by a Borel action of
a countable amenable group is hyperfinite. In this paper we prove that this
question always has a positive answer when the acting group is polycyclic, and
we obtain a positive answer for all free actions of a large class of solvable
groups including the Baumslag--Solitar group BS(1,2) and the lamplighter group.
This marks the first time that a group of exponential volume-growth has been
verified to have this property. In obtaining this result we introduce a new
tool for studying Borel equivalence relations by extending Gromov's notion of
asymptotic dimension to the Borel setting. We show that countable Borel
equivalence relations of finite Borel asymptotic dimension are hyperfinite, and
more generally we prove under a mild compatibility assumption that increasing
unions of such equivalence relations are hyperfinite. As part of our main
theorem, we prove for a large class of solvable groups that all of their free
Borel actions have finite Borel asymptotic dimension (and finite dynamic
asymptotic dimension in the case of a continuous action on a zero-dimensional
space). We also provide applications to Borel chromatic numbers, Borel and
continuous Folner tilings, topological dynamics, and -algebras
Definability and Classification of Equivalence Relations and Logical Theories
This thesis consists of four independent papers.
In the first paper, joint with Kechris, we study the global aspects of structurability in the theory of countable Borel equivalence relations. For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We show that K-structurability interacts well with various preorders commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K; in particular, we characterize the K such that every K-structurable equivalence relation is smooth.
In the second paper, we consider the classes of Kn-structurable equivalence relations, where Kn is the class of n-dimensional contractible simplicial complexes. We show that every Kn-structurable equivalence relation Borel embeds into one structurable by complexes in Kn with the further property that each vertex belongs to at most Mn := 2n-1(n2+3n+2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n=1.
In the third paper, we consider the amalgamation property from model theory in an abstract categorical context. A category is said to have the amalgamation property if every pushout diagram has a cocone. We characterize the finitely generated categories I such that in every category with the amalgamation property, every I-shaped diagram has a cocone.
In the fourth paper, we prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic Lω1ω: every countable Lω1ω-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories (L,T) and (L',T'), every Borel functor Mod(L',T') → Mod(L,T) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L'ω1ω-interpretation of T in T', which generalizes a recent result of Harrison-Trainor, Miller, and Montalban in the case where T, T' are ℵ0-categorical.</p
Essential countability of treeable equivalence relations
We establish a dichotomy theorem characterizing the circumstances under which
a treeable Borel equivalence relation E is essentially countable. Under
additional topological assumptions on the treeing, we in fact show that E is
essentially countable if and only if there is no continuous embedding of E1
into E. Our techniques also yield the first classical proof of the analogous
result for hypersmooth equivalence relations, and allow us to show that up to
continuous Kakutani embeddability, there is a minimum Borel function which is
not essentially countable-to-one
Uniformity, Universality, and Computability Theory
We prove a number of results motivated by global questions of uniformity in
computability theory, and universality of countable Borel equivalence
relations. Our main technical tool is a game for constructing functions on free
products of countable groups.
We begin by investigating the notion of uniform universality, first proposed
by Montalb\'an, Reimann and Slaman. This notion is a strengthened form of a
countable Borel equivalence relation being universal, which we conjecture is
equivalent to the usual notion. With this additional uniformity hypothesis, we
can answer many questions concerning how countable groups, probability
measures, the subset relation, and increasing unions interact with
universality. For many natural classes of countable Borel equivalence
relations, we can also classify exactly which are uniformly universal.
We also show the existence of refinements of Martin's ultrafilter on Turing
invariant Borel sets to the invariant Borel sets of equivalence relations that
are much finer than Turing equivalence. For example, we construct such an
ultrafilter for the orbit equivalence relation of the shift action of the free
group on countably many generators. These ultrafilters imply a number of
structural properties for these equivalence relations.Comment: 61 Page
- …