arXiv subjects
Publications and source records attributed to Dekui Peng.
A topological group is functionally balanced if every bounded real-valued left uniformly continuous function is right uniformly continuous. We prove that every Hausdorff functionally balanced group has coinciding left and right uniformities, giving an affirmative answer to the Itzkowitz problem. Consequently, the classes of SIN, SFSIN and FSIN groups coincide.
We prove that every countable left uniformly discrete subset of a Hausdorff functionally balanced topological group is right thin. As applications, we answer two questions of Bouziad and Troallic: every left precompact subset of a Hausdorff functionally balanced group is right precompact, and every Hausdorff $\omega$-narrow functionally balanced group is a SIN group.
This paper completely classifies when the pro-$p$ completion of an amalgamated product has bounded generation, assuming its starting vertex groups are already boundedly generated. We first prove that the pro-$p$ completion of any abstract amalgam always naturally forms a proper pro-$p$ amalgam. Within this framework, we show that bounded generation is extremely rare. Specifically, the completed group is boundedly generated if and only if one of the vertex groups completely collapses into the edge group (meaning it has an index of 1), or in the highly specific case where $p=2$ and both vertex groups have an index of exactly 2 over the edge intersection.Additionally, we develop a new set of criteria to determine when the original vertex and edge groups map into the completion without collapsing. Finally, we provide a cautionary counterexample: for any given prime $p$, we construct an amalgam whose pro-$p$ completion is boundedly generated, but its full profinite completion is not. This demonstrates that possessing bounded generation at a single prime does not guarantee that the full profinite group will share this property.
We characterize shadowing for continuous endomorphisms of compact Hausdorff groups. First, we prove that an endomorphism has shadowing if and only if its restriction to the identity component does, reducing the problem to compact connected groups. Passing to the stable image then reduces the analysis to surjective endomorphisms. For a compact connected abelian group $A$, let $T_{\mathbb Q}$ be the rational linear endomorphism induced by the dual map on $\widehat A\otimes_{\mathbb Z}\mathbb Q$. We show that the endomorphism of $A$ has shadowing if and only if every finite-dimensional $T_{\mathbb Q}$-invariant subspace is hyperbolic. For a general compact connected group, let $A$ and $S$ denote respectively the connected central and semisimple parts of its stable image. The induced endomorphism on $S/Z(S)$ determines an injective map on the set of simple factors. We prove that shadowing holds exactly when the rational dual map on $A$ is hyperbolic on every finite-dimensional invariant subspace and the induced map on the simple factors has no periodic point.
We study shadowing, with respect to the left uniformity, for continuous endomorphisms of Lie groups and totally disconnected locally compact groups. For Lie groups, an endomorphism has shadowing if and only if its differential is hyperbolic, with zero eigenvalues allowed. This includes singular maps outside the classical theory of Anosov endomorphisms. As consequences, positively expansive Lie group endomorphisms are topologically expanding, while on connected semisimple Lie groups the shadowing endomorphisms are precisely the nilpotent ones. On compact connected Lie groups, the nonsingular case agrees with classical Anosov theory. In sharp contrast, every continuous endomorphism of an arbitrary totally disconnected locally compact group has shadowing, without compactness, metrizability or invertibility assumptions; the proof uses Willis' tidy-above decomposition. Consequently, in this category topological expansion and the topologically Anosov property reduce to positive expansiveness and expansiveness, respectively. We also discuss group shifts and revisit Aoki's dense-orbit compactness result without assuming metrizability.
Shifts of finite type (SFTs) play a central role in the theory of shadowing. Good and Meddaugh showed that SFTs serve as basic building blocks in the structural theory of shadowing systems; in particular, every compact metric dynamical system with shadowing is a factor of the inverse limit of an inverse sequence consisting of SFTs. We first show that, for this factor representation alone, neither shadowing nor metrizability is needed: every compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system consisting of SFTs. Thus, being a factor of an SFT inverse limit is not the structural feature genuinely forced by shadowing. What shadowing provides is stronger stability: in the metric case, every compact shadowing system is a factor of the inverse limit of an inverse sequence of SFTs with surjective bonding maps. Hence the associated zero-dimensional extension still has shadowing. We also prove that every compact Hausdorff shadowing system is conjugate to an inverse limit of metrizable shadowing systems with factor bonding maps.
We investigate the local topological structure of non-metrizable topological groups through the lens of Tukey order and cofinal types. Motivated by recent advances in topological groups admitting an $\omega^\omega$-base, we introduce the \emph{fineness index}, denoted $\f(P)$, for arbitrary directed partially ordered sets. This cardinal invariant fundamentally generalizes the bounding number $\mathfrak{b}$ by capturing the exact threshold where a poset evades domination by its countable subsets, thereby establishing a universal lower bound for the character of topological groups with a $P$-base: $\chi(G) \in \{1, \omega\} \cup [fi(P), \text{cof}(P)]$. Furthermore, we resolve a structural problem regarding the exact cofinal types of free topological groups over uniform spaces. While classical results by Nickolas, Tkachenko, and others successfully computed the character of these groups via cardinal equalities (e.g., $\chi(F(X, \mathcal{U})) = \text{cof}(\mathcal{U}^\omega)$), lifting these equalities to strict Tukey equivalences has remained a persistent combinatorial challenge. By developing the novel machinery of \emph{neat trees} to refine uniform covering trees, we overcome the structural obstructions and prove the Tukey equivalence $\Ne_e(F(X, \U))=_T \U^\omega$ for any compact uniform space $(X, \U)$.
A non-trivial topological group is called \emph{$d$-independent} if for every subgroup of cardinality less than the continuum there exists a countable dense subgroup intersecting it trivially. This notion was introduced by M\'arquez and Tkachenko and has been intensively studied in the metrizable setting. In particular, they proved that a second-countable locally compact abelian group is $d$-independent if and only if it is algebraically an $M$-group, and asked whether the same conclusion holds for all separable locally compact groups. In this paper we give an affirmative answer to this question. We show that every separable locally compact abelian $M$-group is $d$-independent, thereby removing the metrizability assumption from the result of M\'arquez and Tkachenko. In addition, we investigate several further aspects of $d$-independence. We study its behaviour under taking powers of topological groups and extend the notion of $d$-independence to the non-abelian setting. Moreover, we prove that every separable connected compact group is $d$-independent, thereby answering another question posed by M\'arquez and Tkachenko.
Generalizing results from \cite{DTk,DU} we study the fine structure of locally minimal (locally) precompact Abelian groups (these are the locally essential subgroups $G$ of LCA groups $L$, i.e., such that $G$ non-trivially meets all ``small" closed subgroup of $L$). More precisely we prove that if $G$ is a dense locally minimal and sequentially closed subgroup of a LCA group $L$, then the connected component $c(G)$ of $G$ has the same weight as $c(L)$. Moreover, when $w(c(G))$ is not Ulam measurable, then $c(G) = c(L)$. We provide an extended discussion illustrating how this result fails in various ways in the non-abelian case (even for nilpotent groups of class 2). Motivated by the above result, we study further those locally minimal precompact Abelian groups $G$, termed {\em critical locally minimal},such that $c(G) =c(K)$ (where $K$ is the compact completion of $G$) and $G/c(G)$ is not locally minimal. Such a group cannot be compact, neither connected, nor totally disconnected. We provide a proper class of critical locally minimal groups with additional compactness-like properties and we study the class $\CCC$ of compact Abelian groups with a dense critical locally minimal subgroup. In particular, we completely describe the connected components of the finite-dimensional groups belonging to $\CCC$.
We study density spectra associated with dense and closed subgroups of topological groups. For a topological group $G$, let $\dd^*(G)$ denote the set of densities of its dense subgroups. We prove that every compact group satisfies $\dd^*(G)=[d(G),w(G)]$. We also investigate the density spectrum $\cd(G)$ of infinite closed subgroups. In $\mathbf{ZFC}$, we construct a separable countably compact Boolean group containing a closed non-separable subgroup, answering a question of Leiderman, Morris, and Tkachenko. We further show that a free pro-$p$ group of uncountable rank has no non-trivial metrizable closed normal subgroup, giving a negative answer to a problem of Hern\'andez, Hofmann, and Morris. Finally, we study when $\cd(G)$ is an interval. Among other results, we show that under Shelah's Strong Hypothesis ($\mathbf{SSH}$), the closed density spectrum of every infinite $\omega$-bounded group of countable tightness is an interval of cardinals.
Let $\tau$ be an uncountable cardinal. The notion of a \emph{$\tau$-fine} topological group was introduced in 2021. More recently, H. Zhang et al. generalized this concept by defining pseudo-$\tau$-fine topological groups to study certain factorization properties of continuous functions on topological groups. It is known that $\tau$-fineness cannot coexist with precompactness in topological groups with uncountable character. In this paper, we investigate this problem further. We prove that, in topological groups with uncountable pseudocharacter, precompactness can coexist with pseudo-$\tau$-fineness for some bounded $\tau$ but pseudocompactness can never.
Let $G$ be a group and $\sigma, \tau$ be topological group topologies on $G$. We say that $\sigma$ is a successor of $\tau$ if $\sigma$ is strictly finer than $\tau$ and there is not a group topology properly between them. In this note, we explore the existence of successor topologies in topological groups, particularly focusing on non-abelian connected locally compact groups. Our main contributions are twofold: for a connected locally compact group $(G, \tau)$, we show that (1) if $(G, \tau)$ is compact, then $\tau$ has a precompact successor if and only if there exists a discontinuous homomorphism from $G$ into a simple connected compact group with dense image, and (2) if $G$ is solvable, then $\tau$ has no successors. Our work relies on the previous characterization of locally compact group topologies on abelian groups processing successors.
For an infinite group $G$, the poset $\mathcal{L}_G$ of group topologies constitutes a complete lattice. Although $\mathcal{L}_G$ is modular when $G$ is abelian, this property fails to persist for nilpotent groups. Extending Arnautov's 2010 work on the semi-modularity of $\mathcal{L}_G$ for nilpotent groups, we present an alternative proof with enhanced structural clarity. Additionally, we resolve two open questions from the Kourovka Notebook regarding lattice-theoretic properties of $\mathcal{L}_G$: (1) explicit construction of a countably infinite non-abelian nilpotent group with modular topology lattice, and (2) establishing the absence of property $P_2$ in infinite abelian groups.
By [6], a minimal group $G$ is called $z$-minimal if $G/Z(G)$ is minimal. In this paper, we present the $z$-Minimality Criterion for dense subgroups with some applications to topological matrix groups. For a locally compact group $G$, let $\operatorname{Inn}(G)$ be the group of all inner automorphisms of $G,$ endowed with the Birkhoff topology. Using a theorem by Goto [14], we obtain our main result which asserts that if $G$ is a connected Lie group and $H\in\{G/Z(G), \operatorname{Inn}(G)\},$ then $H$ is minimal if and only if it is centre-free and topologically isomorphic to $\operatorname{Inn}(G/Z(G)).$ In particular, if $G$ is a connected Lie group with discrete centre, then $\operatorname{Inn}(G)$ is minimal. We prove that a connected locally compact nilpotent group is $z$-minimal if and only if it is compact abelian. In contrast, we show that there exists a connected metabelian $z$-minimal Lie group that is neither compact nor abelian.
By a recent result of Juh\'{a}sz and van Mill, a locally compact topological group whose dense subspaces are all separable is metrizable. In this note we investigate the following question: is every locally compact group having all dense subgroups separable also metrizable? We give an example to show the answer is negative for locally compact abelian groups, thereby showing that one cannot directly generalize the assertion by replacing ``subspaces'' with ``subgroups''. On the other hand, we prove that the answer is positive for compact groups which are either connected or algebraically abelian; and for locally compact groups containing only separable subgroups. As an application, we obtain a necessary condition for metrizability of pronilpotent groups.
The topological group version of the celebrated Banach-Mazur problem asks wether every infinite topological group has a non-trivial separable quotient group. It is known that compact groups have infinite separable metrizable quotient groups. However, as dense subgroups of compact groups, precompact groups may admit no non-trivial metrizable quotient groups, so also no non-trivial separable quotient groups. In this paper, we study the least cardinal $\mathfrak{m}$ (resp. $\mathfrak{n}$) such that every infinite precompact abelian group admits a quotient group with density character $\leq \mathfrak{m}$ (resp. with weight $\leq \mathfrak{n}$). It is shown that if $2^{<\mathfrak{c}}=\mathfrak{c}$, then $\mathfrak{m}=\mathfrak{c}$ and $\mathfrak{n}=2^\mathfrak{c}$. A more general problem is to describe the set $QW(G)$ of all possible weights of infinite proper quotient groups of a precompact abelian group $G$. We prove that for every subset $E$ of the interval $[\omega, \mathfrak{c}]$, there exists a precompact abelian group $G$ with $QW(G)=E$. If $\omega\in E$, then $G$ can be chosen to be pseudocompact. In an appendix, we give an example to show that a non-totally disconnected locally compact group may admit no separable quotient groups. This answers an open problem posed in \cite{LMT}.
In this note we show that if $G$ is a countably infinite abelian group such that $nG=0$ for some integer $n$, then the only locally minimal group topology on $G$ is the discrete one.