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Longyun Ding

Publications and source records attributed to Longyun Ding.

15 recordsLinked to original sources
arXiv2026

On the complexity of extensions of non-archimedean Polish groups admitting a compatible complete left-invariant metric

In this article, motivated by a problem asked by Allison and Panagiotopoulos, we study a problem concerning the complexity of group extensions within a hierarchy (denoted by $\alpha$-CLI and L-$\alpha$-CLI) on the class of non-archimedean CLI Polish groups: Given a non-archimedean Polish group $G$ and one of its closed normal subgroup $N$, suppose $N$ and $G/N$ are $\alpha$-CLI and $\beta$-CLI, respectively. Is $G$ always $(\alpha+\beta)$-CLI? We provide a positive answer under a certain additional assumption. We then construct two examples yielding negative answers: for each countably infinite ordinal $\alpha$, there exists a group $G$ that is not $\alpha$-CLI, but $G$ has a $1$-CLI normal subgroup $N$ such that $G/N$ is proper $\alpha$-CLI; there exists a proper $3$-CLI group $U$ that has an abelian normal subgroup $N$ such that $U/N$ is also abelian. These examples also provide negative answers to the original problem raised by Allison and Panagiotopoulos. Finally, we show that if $N$ and $G/N$ are $\alpha$-CLI and $\beta$-CLI with $\beta>0$, respectively, then $G$ is $\beta\cdot(\omega\cdot\alpha+1)$-CLI, which gives an upper bound on the complexity of the extended group.

math.LO↗
arXiv2024

On equivalence relations induced by Polish groups admitting compatible two-sided invariant metrics

Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^\omega/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. We first established two results: (1) Let $G,H$ be two Polish groups. If $H$ is TSI but $G$ is not, then $E(G)\not\le_BE(H)$. (2) Let $G$ be a Polish group. Then the following are equivalent: (a) $G$ is TSI non-archimedean; (b)$E(G)\leq_B E_0^\omega$; and (c) $E(G)\leq_B{\mathbb R}^\omega/c_0$. In particular, $E(G)\sim_B E_0^\omega$ iff $G$ is TSI uncountable non-archimedean. A critical theorem presented in this article is as follows: Let $G$ be a TSI Polish group, and let $H$ be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If $E(G)\le_BE(H)$, then there exists a continuous homomorphism $S:G_0\rightarrow H$ such that $\ker(S)$ is non-archimedean, where $G_0$ is the connected component of the identity of $G$. The converse holds if $G$ is connected, $S(G)$ is closed in $H$, and the interval $[0,1]$ can be embedded into $H$. As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fr\'echet spaces, respectively.

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arXiv2023

A hierarchy on non-archimedean Polish groups admitting a compatible complete left-invariant metric

In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by $\alpha$-CLI and L-$\alpha$-CLI where $\alpha$ is a countable ordinal. We establish three results: \begin{enumerate} \item $G$ is $0$-CLI iff $G=\{1_G\}$; \item $G$ is $1$-CLI iff $G$ admits a compatible complete two-sided invariant metric; and \item $G$ is L-$\alpha$-CLI iff $G$ is locally $\alpha$-CLI, i.e., $G$ contains an open subgroup that is $\alpha$-CLI. \end{enumerate} Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups $G_\alpha$ and $H_\alpha$ for $\alpha<\omega_1$, such that \begin{enumerate} \item $H_\alpha$ is $\alpha$-CLI but not L-$\beta$-CLI for $\beta<\alpha$; and \item $G_\alpha$ is $(\alpha+1)$-CLI but not L-$\alpha$-CLI. \end{enumerate}

math.LO↗
arXiv2022

On equivalence relations induced by locally compact abelian Polish groups

Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^\omega/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. The connected component of the identity of a Polish group $G$ is denoted by $G_0$. Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq_B E(H)$, then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker(S)$ is non-archimedean. The converse is also true when $G$ is connected and compact. For $n\in{\mathbb N}^+$, the partially ordered set $P(\omega)/\mbox{Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb R}^n)$ and $E({\mathbb T}^n)$.

math.LO↗
arXiv2022

On equivalence relations induced by Polish groups

The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^\omega/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. Let $G$ be a Polish group. (1) $G$ is a discrete countable group containing at least two elements iff $E(G)\sim_BE_0$; (2) if $G$ is TSI uncountable non-archimedean, then $E(G)\sim_BE_0^\omega$; (3) $G$ is non-archimedean iff $E(G)\le_B=^+$; (4) if $H$ is a CLI Polish group but $G$ is not, then $E(G)\not\le_BE(H)$; (5) if $H$ is a non-archimedean Polish group but $G$ is not, then $E(G)\not\le_BE(H)$. The notion of $\alpha$-l.m.-unbalanced Polish group for $\alpha<\omega_1$ is introduced. Let $G,H$ be Polish groups, $0<\alpha<\omega_1$. If $G$ is $\alpha$-l.m.-unbalanced but $H$ is not, then $E(G)\not\le_B E(H)$. For TSI Polish groups, the existence of Borel reduction is transformed into the existence of a well-behaved continuous mapping between topological groups. As its applications, for any Polish group $G$, let $G_0$ be the connected component of the identity element $1_G$. Let $G$ and $H$ be two separable TSI Lie groups. If $E(G)\le_BE(H)$, then there exists a continuous locally injective map $S:G_0\to H_0$. Moreover, if $G_0,H_0$ are abelian, $S$ is a group homomorphism. In particular, for $c_0,e_0,c_1,e_1\in{\mathbb N}$, $E({\mathbb R}^{c_0}\times{\mathbb T}^{e_0})\le_BE({\mathbb R}^{c_1}\times{\mathbb T}^{e_1})$ iff $e_0\le e_1$ and $c_0+e_0\le c_1+e_1$.

math.LO↗
arXiv2015

Non-Archimedean Abelian Polish Groups and Their Actions

In this paper we consider non-archimedean abelian Polish groups whose orbit equivalence relations are all Borel. Such groups are called tame. We show that a non-archimedean abelian Polish group is tame if and only if it does not involve $\Z^ω$ or $(\Z(p)^{<ω})^ω$ for any prime $p$. In addition to determining the structure of tame groups, we also consider the actions of such groups and study the complexity of their orbit equivalence relations in the Borel reducibility hierarchy. It is shown that if such an orbit equivalence relation is essentially countable, then it must be essentially hyperfinite. We also find an upper bound in the Borel reducibility hierarchy for the orbit equivalence relations of all tame non-archimedean abelian Polish groups.

math.LO↗
arXiv2015

On equivalence relations generated by Schauder bases

In this paper, a notion of Schauder equivalence relation $\mathbb R^\mathbb N/L$ is introduced, where $L$ is a linear subspace of $\mathbb R^\mathbb N$ and the unit vectors form a Schauder basis of $L$. The main theorem is to show that the following conditions are equivalent: (1) the unit vector basis is boundedly complete; (2) $L$ is $F_σ$ in $\mathbb R^\mathbb N$; (3) $\mathbb R^\mathbb N/L$ is Borel reducible to $\mathbb R^\mathbb N/\ell_\infty$. We show that any Schauder equivalence relation generalized by basis of $\ell_2$ is Borel bireducible to $\mathbb R^\mathbb N/\ell_2$ itself, but it is not true for bases of $c_0$ or $\ell_1$. Furthermore, among all Schauder equivalence relations generated by sequences in $c_0$, we find the minimum and the maximum elements with respect to Borel reducibility. We also show that $\mathbb R^\mathbb N/\ell_p$ is Borel reducible to $\mathbb R^\mathbb N/J$ iff $p\le 2$, where $J$ is James' space.

math.LO↗
arXiv2011

On surjectively universal Polish groups

A Polish group is surjectively universal if it can be continuously homomorphically mapped onto every Polish group. Making use of a type of new metrics on free groups \cite{DG}, we prove the existence of surjectively universal Polish groups, answering in the positive a question of Kechris. In fact, we give several examples of surjectively universal Polish groups. We find a sufficient condition to guarantee that the new metrics on free groups can be computed directly. We also compare this condition with CLI groups.

math.LO↗
arXiv2010

Borel reducibility and finitely Holder(α) embeddability

Let $(X_n,d_n),\,n\in\Bbb N$ be a sequence of pseudo-metric spaces, $p\ge 1$. For $x,y\in\prod_{n\in\Bbb N}X_n$, let $(x,y)\in E((X_n)_{n\in\Bbb N};p)\Leftrightarrow\sum_{n\in\Bbb N}d_n(x(n),y(n))^p<+\infty$. For Borel reducibility between equivalence relations $E((X_n)_{n\in\Bbb N};p)$, we show it is closely related to finitely Hölder($α$) embeddability between pseudo-metric spaces.

math.LO↗
arXiv2010

Characterization of $\ell_p$-like and $c_0$-like equivalence relations

Let $X$ be a Polish space, $d$ a pseudo-metric on $X$. If $\{(u,v):d(u,v)<δ\}$ is ${\bfΠ}^1_1$ for each $δ>0$, we show that either $(X,d)$ is separable or there are $δ>0$ and a perfect set $C\subseteq X$ such that $d(u,v)\geδ$ for distinct $u,v\in C$. Granting this dichotomy, we characterize the positions of $\ell_p$-like and $c_0$-like equivalence relations in the Borel reducibility hierarchy.

math.LO↗
arXiv2010

A trichotomy for a class of equivalence relations

Let $X_n, n\in\Bbb N$ be a sequence of non-empty sets, $ψ_n:X_n^2\to\Bbb R^+$. We consider the relation $E((X_n,ψ_n)_{n\in\Bbb N})$ on $\prod_{n\in\Bbb N}X_n$ by $(x,y)\in E((X_n,ψ_n)_{n\in\Bbb N})\Leftrightarrow\sum_{n\in\Bbb N}ψ_n(x(n),y(n))<+\infty$. If $E((X_n,ψ_n)_{n\in\Bbb N})$ is a Borel equivalence relation, we show a trichotomy that either $\Bbb R^\Bbb N/\ell_1\le_B E$, $E_1\le_B E$, or $E\le_B E_0$. We also prove that, for a rather general case, $E((X_n,ψ_n)_{n\in\Bbb N})$ is an equivalence relation iff it is an $\ell_p$-like equivalence relation.

math.LO↗
arXiv2009

Borel reducibility and Holder($α$) embeddability between Banach spaces

We investigate Borel reducibility between equivalence relations $E(X,p)=X^{\Bbb N}/\ell_p(X)$'s where $X$ is a separable Banach space. We show that this reducibility is related to the so called Hölder$(α)$ embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between $E(L_r,p)$'s and $E(c_0,p)$'s for $r,p\in[1,+\infty)$. We also answer a problem presented by Kanovei in the affirmative by showing that $C({\Bbb R}^+)/C_0({\Bbb R}^+)$ is Borel bireducible to ${\Bbb R}^{\Bbb N}/c_0$.

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