Non-existence of universals for classes like reduced torsion free abelian groups under embeddings which are not necessarily pure

Kulikov's problem on universal torsion-free abelian groups

Let T be an abelian group and lambda an uncountable regular cardinal. We consider the question of whether there is a lambda-universal group G^* among all torsion-free abelian groups G of cardinality less than or equal to lambda satisfying Ext(G,T)=0. Here G^* is said to be lambda-universal for T if, whenever a torsion-free abelian group G of cardinality less than or equal to lambda satisfies Ext(G,T)=0, then there is an embedding of G into G^*. For large classes of abelian groups T and cardinals lambda it is shown that the answer is consistently no. In particular, for T torsion, this solves a …

Torsion-Free Abelian Groups are Borel Complete

We prove that the Borel space of torsion-free Abelian groups with domain $ω$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.

The complexity of the embeddability relation between torsion-free abelian groups of uncountable size

We prove that for every uncountable cardinal $κ$ such that $κ^{<κ}=κ$, the quasi-order of embeddability on the $κ$-space of $κ$-sized graphs Borel reduces to the embeddability on the $κ$-space of $κ$-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability on the $κ$-space of $κ$-sized $R$-modules, for every $\mathbb{S}$-cotorsion-free ring $R$ of cardinality less than the continuum. As a consequence we get that all the previous are complete $\boldsymbolΣ^1_1$ quasi-orders.

Torsion-Free Abelian Groups are Consistently $a Δ^1_2$-complete

Let $\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $ZFC^- + κ(ω)$ exists, then $\mbox{TFAG}$ is $a Δ^1_2$-complete; in particular, this is consistent with $ZFC$. We define the $α$-ary Schröder- Bernstein property, and show that $\mbox{TFAG}$ fails the $α$-ary Schröder-Bernstein property for every $α< κ(ω)$. We leave open whether or not $\mbox{TFAG}$ can have the $κ(ω)$-ary Schröder-Bernstein property; if it did, then it would not be $a Δ^1_2$-complete, and hence not Borel complete.

A countable universal torsion abelian group for purity

We show that there is a countable universal abelian p-group for purity, i.e., a countable abelian p-group $U$ such that every countable abelian p-group purely embeds in $U$. This is the last result needed to provide a complete solution to Problem 5.1 of [Fuc15] below $\aleph_ω$. We introduce $\aleph_0$-strongly homogeneous p-groups, show that there is a universal abelian p-group for purity which is $\aleph_0$-strongly homogeneous, and completely characterize the countable $\aleph_0$-strongly homogeneous p-groups.

无 torsion 的 Abel 群是忠实 Borel 完整的，并且纯可嵌入性是一个完全的解析准序

Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order

本文证明了无 torsion 的 Abel 群是忠实 Borel 完整的，并展示了纯可嵌入性是一个完全的解析准序。

Borel reductions of profinite actions of SL(n,Z)

Greg Hjorth and Simon Thomas proved that the classification problem for torsion-free abelian groups of finite rank \emph{strictly increases} in complexity with the rank. Subsequently, Thomas proved that the complexity of the classification problems for $p$-local torsion-free abelian groups of fixed rank $n$ are \emph{pairwise incomparable} as $p$ varies. We prove that if $3\leq m<n$ and $p,q$ are distinct primes, then the complexity of the classification problem for $p$-local torsion-free abelian groups of rank $m$ is again incomparable with that for $q$-local torsion-free abelian groups of ra…

Some ugly aleph_1-free abelian groups

Given an aleph_1-free abelian group G we characterize the class C_G of all torsion abelian groups T satisfying Ext(G,T)=0 assuming the continuum hypothesis CH. Moreover, in Godel's constructable universe we prove that this characterizes C_G for arbitrary torsion-free abelian G. It follows that there exist some ugly aleph_1-free abelian groups.

On Holder-Brascamp-Lieb inequalities for torsion-free discrete Abelian groups

Hölder-Brascamp-Lieb inequalities provide upper bounds for a class of multilinear expressions, in terms of $L^p$ norms of the functions involved. They have been extensively studied for functions defined on Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for discrete Abelian groups and, in terms of suitable data, have characterized the set of all tuples of exponents for which such an inequality holds for specified data, as the convex polyhedron defined by a particular finite set of affine inequalities. In this paper we advance the theory of such inequ…

无挠阿贝尔群扩张的分类问题，I

The classification problem for extensions of torsion-free abelian groups, I

本文在Borel可定义同调代数框架下，确定了无挠可数阿贝尔群被特定类型阿贝尔群扩张的分类复杂度，并得出扩张分裂的有限秩子群判别准则。

ZFC中的华莱士问题与可数紧无挠阿贝尔群

The Wallace problem and countably compact torsion-free Abelian groups in ZFC

该研究在ZFC公理体系下，证明基数为$\mathfrak c$的无挠阿贝尔群可赋予无非平凡收敛序列的豪斯多夫可数紧群拓扑，构造出满足双侧消去律但非群的可数紧拓扑半群，否定回答了华莱士问题，并衍生出相关拓扑结构结论。

非阿贝尔dp-极小群I：无挠与远隔情形

On non-abelian dp-minimal groups I: the torsion-free and distal cases

本文解决具有远隔f-泛型类型的dp-极小群是否为幂零-有限群的问题，证明此类群是幂零-有限群，还给出无挠dp-极小群必为阿贝尔群等相关结果。