256 research outputs found
On the complexity of the relations of isomorphism and bi-embeddability
Given an L_{\omega_1 \omega}-elementary class C, that is the collection of
the countable models of some L_{\omega_1 \omega}-sentence, denote by \cong_C
and \equiv_C the analytic equivalence relations of, respectively, isomorphism
and bi-embeddability on C. Generalizing some questions of Louveau and Rosendal
[LR05], in [FMR09] it was proposed the problem of determining which pairs of
analytic equivalence relations (E,F) can be realized (up to Borel
bireducibility) as pairs of the form (\cong_C,\equiv_C), C some L_{\omega_1
\omega}-elementary class (together with a partial answer for some specific
cases). Here we will provide an almost complete solution to such problem: under
very mild conditions on E and F, it is always possible to find such an
L_{\omega_1 \omega}-elementary class C.Comment: 15 page
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 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
Invariantly universal analytic quasi-orders
We introduce the notion of an invariantly universal pair (S,E) where S is an
analytic quasi-order and E \subseteq S is an analytic equivalence relation.
This means that for any analytic quasi-order R there is a Borel set B invariant
under E such that R is Borel bireducible with the restriction of S to B. We
prove a general result giving a sufficient condition for invariant
universality, and we demonstrate several applications of this theorem by
showing that the phenomenon of invariant universality is widespread. In fact it
occurs for a great number of complete analytic quasi-orders, arising in
different areas of mathematics, when they are paired with natural equivalence
relations.Comment: 31 pages, 1 figure, to appear in Transactions of the American
Mathematical Societ
The complexity of classifying separable Banach spaces up to isomorphism
It is proved that the relation of isomorphism between separable Banach spaces
is a complete analytic equivalence relation, i.e., that any analytic
equivalence relation Borel reduces to it. Thus, separable Banach spaces up to
isomorphism provide complete invariants for a great number of mathematical
structures up to their corresponding notion of isomorphism. The same is shown
to hold for (1) complete separable metric spaces up to uniform homeomorphism,
(2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to
(complemented) biembeddability, (4) Polish groups up to topological
isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the
constructions rely on methods recently developed by S. Argyros and P. Dodos
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
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