arXiv subjects

Menachem Shlossberg

Publications and source records attributed to Menachem Shlossberg.

At least 19 recordsLinked to original sources

arXiv2026

The Addition Theorem for the Algebraic Entropy of Nilpotent Groups

The Addition Theorem for the algebraic entropy of group endomorphisms was first proved for torsion abelian groups by Dikranjan, Goldsmith, Salce and Zanardo, and was later extended to all abelian groups by Dikranjan and Giordano Bruno. Shlossberg subsequently proved it for torsion nilpotent groups of class 2. As our main result, we prove the Addition Theorem for endomorphisms of nilpotent groups of arbitrary nilpotency class, without any torsion assumption. In the torsion case, this yields in particular that, if G is a torsion nilpotent group and \phi\in\operatorname{End}(G), then either h(\phi)=\infty or h(\phi)=\log(\alpha) for some positive integer \alpha. We further prove that, for every group G and every automorphism \phi\in\operatorname{Aut}(G), the Addition Theorem holds with respect to each term of the upper central series. As a consequence, the Addition Theorem holds for automorphisms of every \omega-hypercentral group. Finally, we establish a reduction principle: if \mathfrak X is a class of locally finite groups closed under taking subgroups and quotients, then the Addition Theorem holds for endomorphisms of groups in \mathfrak X if and only if it holds for endomorphisms of those groups in \mathfrak X generated by bounded subsets, that is, subsets whose element orders are uniformly bounded.

math.GR↗

arXiv2024

Key subgroups in topological groups

We introduce two minimality properties of subgroups in topological groups. A subgroup $H$ is a key subgroup (co-key subgroup) of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$ (resp., on the coset space $G/H$) the original topology. Every co-minimal subgroup is a key subgroup while the converse is not true. Every locally compact co-compact subgroup is a key subgroup (but not always co-minimal). Any relatively minimal subgroup is a co-key subgroup (but not vice versa). Extending some results concerning the generalized Heisenberg groups, we prove that the center ("corner" subgroup) of the upper unitriangular group $\mathrm{UT(n,K)}$, defined over a commutative topological unital ring $K$, is a key subgroup. Every "non-corner" 1-parameter subgroup $H$ of $\mathrm{UT(n,K)}$ is a co-key subgroup. We study injectivity property of the restriction map $$r_H \colon \mathcal{T}_{\downarrow}(G) \to \mathcal{T}_{\downarrow}(H), \ σ\mapsto σ|_H$$ and show that it is an isomorphism of sup-semilattices for every central co-minimal subgroup $H$, where $\mathcal{T}_{\downarrow}(G)$ is the semilattice of coarser Hausdorff group topologies on $G$.

math.GN↗

arXiv2024

Minimality of the inner automorphism group

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.

math.GN↗

arXiv2024

Non-archimedean topological monoids

We say that a topological monoid $S$ is left non-archimedean (in short: l-NA) if the left action of $S$ on itself admits a proper $S$-compactification $ν\colon S \hookrightarrow Y$ such that $Y$ is a Stone space. This provides a natural generalization of the well known concept of NA topological groups. The Stone and Pontryagin dualities play major role in achieving useful characterizations of NA monoids. We discuss universal NA monoids and show that many naturally defined topological monoids are NA. We show that many naturally defined topological monoids are NA and present universal NA monoids. Among others, we prove that the Polish monoid $C(2^ω,2^ω)$ is a universal separable metrizable l-NA monoid and the Polish monoid ${\mathbb N}^{\mathbb N}$ is universal for separable metrizable r-NA monoids.

math.GN↗

arXiv2022

Minimality conditions equivalent to the finitude of Fermat and Mersenne primes

It is still open whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning the Mersenne numbers is also unsolved. Extending some results from [9], we characterizethe the Fermat primes and the Mersenne primes in terms of topological minimality of some matrix groups. This is done by showing, among other things, that if $\Bbb{F}$ is a subfield of a local field of characteristic $\neq 2,$ then the special upper triangular group $\operatorname{ST^+}(n,\Bbb{F})$ is minimal precisely when the special linear group $\operatorname{SL}(n,\Bbb{F})$ is. We provide criteria for the minimality (and total minimality) of $\operatorname{SL}(n,\Bbb{F})$ and $\operatorname{ST^+}(n,\Bbb{F}),$ where $\Bbb{F}$ is a subfield of $\Bbb{C}.$ Let $\mathcal F_π$ and $\mathcal F_c $ be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for $\mathcal{A}\in \{\mathcal F_π, \mathcal F_c\}:$ $\bullet \ \mathcal{A}$ is finite; $\bullet \ \prod_{F_n\in \mathcal{A}}\operatorname{SL}(F_n-1, \Bbb{Q}(i))$ is minimal, where $\Bbb{Q}(i)$ is the Gaussian rational field; $\bullet \ \prod_{F_n\in \mathcal{A}}\operatorname{ST^+}(F_n-1, \Bbb{Q}(i))$ is minimal. Similarly, denote by $\mathcal M_π$ and $\mathcal M_c $ the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let $\mathcal{B}\in\{ \mathcal M_π, \mathcal M_c\}.$ Then the following conditions are equivalent: $\bullet \ \mathcal B$ is finite; $\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{SL}(M_p+1, \Bbb{Q}(i))$ is minimal; $\bullet \ \prod_{M_p\in \mathcal{B}}\operatorname{ST^+}(M_p+1, \Bbb{Q}(i))$ is minimal.

math.GN↗

arXiv2022

Minimality of topological matrix groups and Fermat primes

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group $SUT(n,\mathbb{F})$ is minimal for every local field $\mathbb{F}$ of characteristic $\neq 2$. This result is new even for the field $\mathbb{R}$ of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group $SL(n,\mathbb{F})$, where $\mathbb{F}$ is a subfield of a local field. This extends some known results of Remus-Stoyanov (1991) and Bader-Gelander (2017). One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime $p$ the following conditions are equivalent: $\bullet$ $p$ is a Fermat prime; $\bullet$ $SL(p-1,\mathbb{Q})$ is minimal, where $\mathbb{Q}$ is the field of rationals equipped with the $p$-adic topology; $\bullet$ $SL(p-1,\mathbb{Q}(i))$ is minimal, where $\mathbb{Q}(i) \subset \mathbb{C}$ is the Gaussian rational field.

math.GN↗

arXiv2022

The Addition theorem for two-step nilpotent torsion groups

The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [4]. Later, this result was extended to all abelian groups [3] and, recently, to all torsion finitely quasihamiltonian groups [7]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [8]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.

math.GR↗

arXiv2021

C-Minimal topological groups

We study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of Hall and Kulatilaka and a characterization of a certain class of Lie groups, due to Grosser and Herfort, we prove that a c-minimal locally solvable Lie group is compact. It is shown that if a topological group $G$ contains a compact open normal subgroup $N$, then $G$ is c-totally minimal if and only if $G/N$ is hereditarily non-topologizable. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering a question by Dikranjan and Megrelishvili we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact. We also prove that the group $A\times F$ is c-(totally) minimal for every (respectively, totally) minimal abelian group $A$ and every finite group $F.$

math.GN↗

arXiv2019

Algebraic entropy on topologically quasihamiltonian groups

We study the algebraic entropy of continuous endomorphisms of compactly covered, locally compact, topologically quasihamiltonian groups. We provide a Limit-free formula which helps us to simplify the computations of this entropy. Moreover, several Addition Theorems are given. In particular, we prove that the Addition Theorem holds for endomorphisms of quasihamiltonian torsion FC-groups (e.g., Hamiltonian groups).

math.DS↗

arXiv2018

Densely locally minimal groups

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups $\mathbb Z_p$ of $p$-adic integers. In [31], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups. We prove that in case that a topological abelian group $G$ is either compact or connected locally compact, then $G$ is densely locally minimal if and only if $G$ either is a Lie group or has an open subgroup isomorphic to $\mathbb Z_p$ for some prime $p$. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to $\mathbb Z_p$. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to $\mathbb Z_p$.

math.GN↗

arXiv2018

Algebraic entropy on strongly compactly covered groups

We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups $G$ such that every element of $G$ is contained in a compact open normal subgroup of $G$. For continuous endomorphisms $ϕ:G\to G$ of these groups we compute the algebraic entropy and study its properties. Also an Addition Theorem is available under suitable conditions.

math.GN↗

arXiv2018

Hereditarily minimal topological groups

We study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal. In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups $\mathbb Z_p$ of $p$-adic integers. We extend Prodanov's theorem to the non-abelian case at several levels. For infinite hypercentral (in particular, nilpotent) locally compact groups we show that the hereditarily minimal ones remain the same as in the abelian case. On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that in particular they are always compact and metabelian. The proofs involve the (hereditarily) locally minimal groups, introduced similarly. In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group $\mathbb Q_p\rtimes \mathbb Q_p^*$ is hereditarily locally minimal, where $\mathbb Q_p^*$ is the multiplicative group of non-zero $p$-adic numbers acting on the first component by multiplication. Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group.

math.GN↗

arXiv2017

Balanced and functionally balanced $P$-groups

In relation to Itzkowitz's problem, we show that a $\mathfrak c$-bounded $P$-group is balanced if and only if it is functionally balanced. We prove that for an arbitrary $P$-group, being functionally balanced is equivalent to being strongly functionally balanced. A special focus is given to the uniform free topological group defined over a uniform $P$-space. In particular, we show that this group is (functionally) balanced precisely when its subsets $B_n,$ consisting of words of length at most $n,$ are all (resp., functionally) balanced.

math.GN↗

arXiv2016

Minimality of the Semidirect Product

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. We provide a sufficient and necessary condition for the minimality of the semidirect product $G\leftthreetimes P,$ where $G$ is a compact topological group and $P$ is a topological subgroup of $Aut(G)$. We prove that $G\leftthreetimes P$ is minimal for every closed subgroup $P$ of $Aut(G)$. In case $G$ is abelian, the same is true for every subgroup $P \subseteq Aut(G)$. We show, in contrast, that there exist a compact two-step nilpotent group $G$ and a subgroup $P$ of $Aut(G)$ such that $G\leftthreetimes P$ is not minimal. This answers a question of Dikranjan. Some of our results were inspired by a work of Gamarnik.

math.GN↗

arXiv2016

Non-archimedean transportation problems and Kantorovich ultra-norms

We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field $\mathbb Q_{p}$ of $p$-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present NA versions of the Arens-Eells construction and of the integer value property. We introduce and study free NA locally convex spaces. In particular, we provide conditions under which these spaces are normable by Kantorovich ultra-norms and also conditions which yield NA versions of Tkachenko-Uspenskij theorem about free abelian topological groups.

math.FA↗

arXiv2014

On Graev type ultra-metrics

We study Graev ultra-metrics which were introduced by Gao. We show that the free non-archimedean balanced topological group defined over an ultra-metric space is metrizable by a Graev ultra-metric. We prove that the Graev ultra-metric has a maximal property. Using this property, among others, we show that the Graev ultra-metric associated with an ultra-metric space $(X,d)$ with diameter$\leq 1$ coincides with the ultra-metric $\hat{d}$ of Savchenko and Zarichnyi.

math.GN↗

arXiv2013

Free non-archimedean topological groups

We study free topological groups defined over uniform spaces in some subclasses of the class NA of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean NA groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another contrasting advantage is that the induced topological group actions on free abelian NA groups frequently remain continuous. One of the main applications is: any epimorphism in the category NA must be dense. Moreover, the same methods improve the following result of T.H. Fay : the inclusion of a proper open subgroup H into G is not an epimorphism in the category of all Hausdorff topological groups. A key tool in the proofs is Pestov's test of epimorphisms. Our results provide a convenient way to produce surjectively universal NA abelian and balanced groups. In particular, we unify and strengthen some recent results of Gao and Gao-Xuan as well as classical results about profinite groups which go back to Iwasawa and Gildenhuys-Lim.

math.GN↗

arXiv2011

Notes on non-archimedean topological groups

We show that the Heisenberg type group $H_X=(\Bbb{Z}_2 \oplus V) \leftthreetimes V^{\ast}$, with the discrete Boolean group $V:=C(X,\Z_2)$, canonically defined by any Stone space $X$, is always minimal. That is, $H_X$ does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean $G$ there exists a (resp., locally compact) non-archimedean minimal group $M$ such that $G$ is a group retract of $M.$ For discrete groups $G$ the latter was proved by S. Dierolf and U. Schwanengel. We unify some old and new characterization results for non-archimedean groups. Among others we show that every continuous group action of $G$ on a Stone space $X$ is a restriction of a continuous group action by automorphisms of $G$ on a topological (even, compact) group $K$. We show also that any epimorphism $f: H \to G$ (in the category of Hausdorff topological groups) into a non-archimedean group $G$ must be dense.

math.GN↗