256 research outputs found

    On the complexity of the relations of isomorphism and bi-embeddability

    Full text link
    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

    Get PDF
    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

    Get PDF
    We prove that for every uncountable cardinal Îș\kappa such that Îș<Îș=Îș\kappa^{<\kappa}=\kappa, the quasi-order of embeddability on the Îș\kappa-space of Îș\kappa-sized graphs Borel reduces to the embeddability on the Îș\kappa-space of Îș\kappa-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability on the Îș\kappa-space of Îș\kappa-sized RR-modules, for every S\mathbb{S}-cotorsion-free ring RR of cardinality less than the continuum. As a consequence we get that all the previous are complete ÎŁ11\boldsymbol{\Sigma}^1_1 quasi-orders.Comment: 14 pages, final versio

    Invariantly universal analytic quasi-orders

    Full text link
    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

    Full text link
    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

    Get PDF
    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
    • 

    corecore