9 research outputs found
Universal countable Borel quasi-orders
In recent years, much work in descriptive set theory has been focused on the
Borel complexity of naturally occurring classification problems, in particular,
the study of countable Borel equivalence relations and their structure under
the quasi-order of Borel reducibility. Following the approach of Louveau and
Rosendal for the study of analytic equivalence relations, we study countable
Borel quasi-orders.
In this paper we are concerned with universal countable Borel quasi-orders,
i.e. countable Borel quasi-orders above all other countable Borel quasi-orders
with regard to Borel reducibility. We first establish that there is a universal
countable Borel quasi-order, and then establish that several countable Borel
quasi-orders are universal. An important example is an embeddability relation
on descriptive set theoretic trees.
Our main result states that embeddability of finitely generated groups is a
universal countable Borel quasi-order, answering a question of Louveau and
Rosendal. This immediately implies that biembeddability of finitely generated
groups is a universal countable Borel equivalence relation. The same techniques
are also used to show that embeddability of countable groups is a universal
analytic quasi-order.
Finally, we show that, up to Borel bireducibility, there are continuum-many
distinct countable Borel quasi-orders which symmetrize to a universal countable
Borel equivalence relation
The complexity of the embeddability relation between torsion-free abelian groups of uncountable size
We prove that for every uncountable cardinal such that
, the quasi-order of embeddability on the
-space of -sized graphs Borel reduces to the embeddability on
the -space of -sized torsion-free abelian groups. Then we use
the same techniques to prove that the former Borel reduces to the embeddability
on the -space of -sized -modules, for every
-cotorsion-free ring of cardinality less than the continuum. As
a consequence we get that all the previous are complete
quasi-orders.Comment: 14 pages, final versio
Universality of group embeddability
Working in the framework of Borel reducibility, we study various notions of
embeddability between groups. We prove that the embeddability between countable
groups, the topological embeddability between (discrete) Polish groups, and the
isometric embeddability between separable groups with a bounded bi-invariant
complete metric are all invariantly universal analytic quasi-orders. This
strengthens some results from [Wil14] and [FLR09].Comment: Minor corrections. 15 pages, submitte
The bi-embeddability relation for finitely generated groups II
We study the isomorphism and bi-embeddability relations on the spaces of Kazhdan groups and finitely generated simple groups
Invariant universality for quandles and fields
We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel reducibility states that any analytic quasi-order on a standard Borel space essentially appears as the restriction of the embeddability relation to an isomorphism-invariant Borel set. As an intermediate step we show that the embeddability relation of countable quandles is a complete analytic quasi-order
The bi-embeddability relation for finitely generated groups
There does not exist a Borel selection of an isomorphism class within each bi-embeddability class
of finitely generated groups
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
Countable Borel quasi-orders
In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal in cite{LR05} for the study of analytic equivalence relations, we study countable Borel quasi-orders. We are largely concerned in this thesis with universal countable Borel quasi-orders, i.e. countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We first establish that there is a universal countable Borel quasi-order, using a Feldman-Moore-type result for countable Borel quasi-orders and an argument similar to that of Dougherty, Jackson, and Kechris in cite{DJK94}. We then establish that several countable Borel quasi-orders are universal. An important example is an embeddability relation on descriptive set theoretic trees. This is used in many of the other proofs of universality. Our main result is Theorem 5.5.2, which states that embeddability of finitely generated groups is a universal countable Borel quasi-order, answering a question of Louveau and Rosendal in cite{LR05}. This immediately implies that biembeddability of finitely generated groups is a universal countable Borel equivalence relation. Although it may have been possible to prove this only using results on countable Borel equivalence relations, the use of quasi-orders seems to be the most direct route to this result. The proof uses small cancellation theory. The same techniques are also used to show that embeddability of countable groups is a universal analytic quasi-order. Finally, we discuss the structure of countable Borel quasi-orders under Borel reducibility, and we present some open problems.Ph. D.Includes bibliographical referencesby Jay William