Papers by Dikran Dikranjan
Journal of Algebra, 2020
We introduce two notions of algebraic entropy for actions of cancellative right amenable semigrou... more We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups S on discrete abelian groups A by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case S = N. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of the Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.
arXiv (Cornell University), Jul 24, 2008
For an abelian topological group G let G^* be the dual group of all continuous characters endowed... more For an abelian topological group G let G^* be the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G) and an open neighbourhood U of 0 in the circle group, we show that the set of all characters which send X into U has the same size as G^*. (Here w(G) denotes the weight of G.) A subgroup D of G determines G if the restriction homomorphism G^* --> D^* of the dual groups is a topological isomorphism. We prove that w(G) = min {|D|: D is a subgroup of G that determines G} for every compact abelian group G. In particular, an infinite compact abelian group determined by its countable subgroup must be metrizable. This gives a negative answer to questions of Comfort, Hernandez, Macario, Raczkowski and Trigos-Arrieta. As an application, we furnish a short elementary proof of the result from [11] that compact determined abelian groups are metrizable.
arXiv (Cornell University), Jul 24, 2008
For an abelian topological group G let G^* be the dual group of all continuous characters endowed... more For an abelian topological group G let G^* be the dual group of all continuous characters endowed with the compact open topology. A subgroup D of G determines G if the restriction homomorphism G^* --> D^* of the dual groups is a topological isomorphism. Given a scattered compact subset X of an infinite compact abelian group G such that |X|<w(G) and an open neighbourhood U of 0 in the circle group, we show that the set of all characters which send X into U has the same size as G^*. (Here w(G) denotes the weight of G.) As an application, we prove that a compact abelian group determined by its countable subgroup must be metrizable. This gives a negative answer to questions of Comfort, Hernandez, Macario, Raczkowski and Trigos-Arrieta, as well as provides short proofs of main results established in three manuscripts by these authors.
arXiv (Cornell University), Dec 15, 2008
Let G be an abelian topological group. The symbol G denotes the group of all continuous character... more Let G be an abelian topological group. The symbol G denotes the group of all continuous characters χ : G → T endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that χ(E) ⊆ ϕ([−1/4, 1/4]) holds only for the trivial character χ ∈ G, where ϕ : R → T = R/Z is the canonical homomorphism. A super-sequence is a non-empty compact Hausdorff space S with at most one non-isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G a contains a super-sequence converging to 0 that is qc-dense in G. This gives as a corollary a recent theorem of Außenhofer: For a connected locally compact abelian group G, the restriction homomorphism r : G → G a defined by r(χ) = χ ↾ Ga for χ ∈ G, is a topological isomorphism. We also show that an infinite compact group G is connected if and only if its arc component G a contains a super-sequence S converging to the identity e that generates a dense subgroup of G (equivalently, S \ {e} is an infinite suitable set for G in the sense of Hofmann and Morris).
Elsevier eBooks, 2007
Publisher Summary This chapter discusses selected topics from the structure theory of topological... more Publisher Summary This chapter discusses selected topics from the structure theory of topological groups. It contains open problems and questions covering the a number of topics including: the dimension theory of topological groups, pseudocompact and countably compact group topologies on Abelian groups, with or without nontrivial convergent sequences, categorically compact groups, sequentially complete groups, the Markov–Zariski topology, the Bohr topology, and transversal group topologies. All topological groups considered in this chapter are assumed to be Hausdorff. It is stated that Abelian group G is algebraically compact provided that an Abelian group H is found such that G × H admits a compact group topology. Algebraically compact groups form a relatively narrow subclass of Abelian groups (for example, the group ℤ of integers is not algebraically compact). On the other hand, every Abelian group G is algebraically pseudo-compact; that is, an Abelian group H can be found such that G × H ∈ P . Problems and questions related to Bohr-homeomorphic bounded Abelian groups are also discussed in the chapter.
The closure operators in the category Top of topological spaces are studied in full detail, provi... more The closure operators in the category Top of topological spaces are studied in full detail, providing a classification via their general properties as idempotency, additivity, (weak) hereditariness, (finite) productivity, etc
Journal of Group Theory, Jan 3, 2002
An Error Occurred Setting Your User Cookie. This site uses cookies to improve performance. If you... more An Error Occurred Setting Your User Cookie. This site uses cookies to improve performance. If your browser does not accept cookies, you cannot view this site. Setting Your Browser to Accept Cookies. There are many reasons why a cookie could not be set correctly. ...
... Categorical Structure of Closure Operators With Applications to Topology, Algebra and Discret... more ... Categorical Structure of Closure Operators With Applications to Topology, Algebra and DiscreteMathematics by D. Dikranjan Department of Mathematics, University of Udine, Udine, Italy and W. Tholen Department of Mathematics and Statistics, York University, North York ...
We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately m... more We introduce the algebraic entropy for endomorphisms of arbitrary abelian groups, appropriately modifying existing notions of entropy. The basic properties of the algebraic entropy are given, as well as various examples. The main result of this paper is the Addition Theorem showing that the algebraic entropy is additive in appropriate sense with respect to invariant subgroups. We give several applications of the Addition Theorem, among them the Uniqueness Theorem for the algebraic entropy in the category of all abelian groups and their endomorphisms. Furthermore, we point out the delicate connection of the algebraic entropy with the Mahler measure and Lehmer Problem in Number Theory.
Journal of Mathematical Analysis and Applications, Jul 1, 2019
Fundamenta Mathematicae, 2020
In this paper, we introduce a new version of characterized subgroups of the circle group T that w... more In this paper, we introduce a new version of characterized subgroups of the circle group T that we call statistically characterized subgroups (briefly, s-characterized subgroups). We primarily investigate these subgroups for arithmetic sequences of integers and show that these subgroups are essentially different and strictly larger in size than the much investigated class of characterized subgroups, having cardinality c but remaining nontrivial (i.e. different from T) though remaining topologically nice
Topology and its Applications, Jun 1, 2019
Given a compact abelian group G, we study the poset G δ (G) of G δ-dense subgroups of G and the s... more Given a compact abelian group G, we study the poset G δ (G) of G δ-dense subgroups of G and the subgroup δ-den(G) := G δ (G), named G δ-density nucleus of G. It turns out that for every compact abelian group G: (i) the subgroup δ-den(G) is compact metrizable and coincides with the intersection of just two members D 0 , D 1 ∈ G δ (G); (ii) if δ-den(G) = {0}, then there exists some independent family F in G δ (G) (i.e., with F = i∈I D i) such that: (ii 1) |F| = r(G), the free rank of G, if mG is not metrizable for any positive m ∈ N; otherwise, (ii 2) when G is (necessarily) bounded torsion, |F| coincides with the least leading Ulm-Kaplansky invariant of G. (iii) in option (ii 1) the members of F can be chosen to be free subgroups of G precisely when G admits a free dense subgroup (equivalently, when r(G) ≥ d(G)); F can have (the maximum possible) size |G| if and only if r(G) = |G|; (iv) in option (ii 2) the family F can have size |G| if and only if all leading Ulm-Kaplansky invariants of G coincide with |G|. Resuming, the compact abelian groups G with trivial G δ-density nucleus δ-den(G) are, roughly speaking, "maximally fragmented" into G δ-dense subgroups. More precisely, the existence of a pair of subgroups D 0 , D 1 ∈ G δ (G) with D 0 ∩ D 1 = {0} yields the existence of a large independent family F = {D i ∈ I} ⊆ G δ (G) with I is as big as possible (i.e., as big as the algebraic structure of G permits in terms of some immediate necessary restraints involving relevant algebraic cardinal invariants of the group, in the spirit of those in (ii)-(iv)).
Topology and its Applications, Mar 1, 2016
Abstract We introduce and study a functorial topology on every group G having as a base the famil... more Abstract We introduce and study a functorial topology on every group G having as a base the family of all subgroups of G . Making use of this topology, we obtain an equivalent description of the small subgroup generating property introduced by Gould [26] ; see also Comfort and Gould [6] . This property implies minimal almost periodicity. Answering questions of Comfort and Gould [6] , we show that every abelian group of infinite divisible rank admits a group topology having the small subgroup generating property. For unbounded abelian groups of finite divisible rank, we find a new necessary condition for the existence of a group topology having the small subgroup generating property, and we conjecture that this condition is also sufficient.
Ricerche Di Matematica, Apr 18, 2018
A subgroup H of the circle group T is said to be characterized by a sequence u = (u n) n∈N of int... more A subgroup H of the circle group T is said to be characterized by a sequence u = (u n) n∈N of integers, if H = {x ∈ T : u n x → 0}. The characterized subgroups of T are known also under the name topologically u-torsion subgroups. This survey paper is dedicated to characterized subgroups of T: we recall their main properties and collect most of the basic results from the wide bibliography, following, when possible, the historical line, and trying to show the deep roots of this topic in several areas of Mathematics. Due to this universality of the topic, many notions and results were found independently by various authors working unaware of each other, so our effort is also directed towards giving credit to all of them to the best of our knowledge. We provide also some background on the notions of characterized subgroup and topologically u-torsion subgroup in the general case of topological abelian groups, where they differ very substantially.
Topology and its Applications, 2023
arXiv (Cornell University), Jun 26, 2010
The algebraic entropy h, defined for endomorphisms φ of abelian groups G, measures the growth of ... more The algebraic entropy h, defined for endomorphisms φ of abelian groups G, measures the growth of the trajectories of non-empty finite subsets F of G with respect to φ. We show that this growth can be either polynomial or exponential. The greatest φ-invariant subgroup of G where this growth is polynomial coincides with the greatest φ-invariant subgroup P(G, φ) of G (named Pinsker subgroup of φ) such that h(φ ↾ P(G,φ)) = 0. We obtain also an alternative characterization of P(G, φ) from the point of view of the quasi-periodic points of φ. This gives the following application in ergodic theory: for every continuous injective endomorphism ψ of a compact abelian group K there exists a largest ψ-invariant closed subgroup N of K such that ψ ↾ N is ergodic; furthermore, the induced endomorphism ψ of the quotient K/N has zero topological entropy.
arXiv (Cornell University), Dec 9, 2014
In a compact abelian group X, a subgroup H is called characterized if there exists a sequence of ... more In a compact abelian group X, a subgroup H is called characterized if there exists a sequence of characters v = (v n) of X such that H = {x ∈ X : v n (x) → 0 in T}. Gabriyelyan proved for X = T, that the characterized subgroup {x ∈ T : n!x → 0 in T} is not an F σ-set. In this paper, we obtain a complete description of the F σ-subgroups of T characterized by sequences of integers v = (v n) such that v n |v n+1 for all n ∈ N showing that these are exactly the countable subgroups of T. Moreover, in the general setting of compact metrizable abelian groups, we give a new point of view to study the Borel complexity of characterized subgroups in terms of appropriate test-topologies defined on the group X. We denote by T = R/Z the additive circle group and by ϕ : R → T is the canonical homomorphism; furthermore, we identify an element x ∈ T with its unique pre-image in [0, 1) under ϕ. We use for the standard norm on the torus; namely, if x ∈ T, then x = x if x < 1 2 , otherwise x = 1 − x. The metric generated by will be denoted by d. For an abelian group G we denote by t(G) the torsion subgroup of G. The closure of a subset M of a topological space will be denoted by M. For a topological abelian group X a character of X is a continuous homomorphism χ : X → T. Denote by X the group of all characters of X, that is the Pontryagin dual of X.
arXiv (Cornell University), May 14, 2019
Coarse geometry is the study of large-scale properties of spaces. In this paper we study group co... more Coarse geometry is the study of large-scale properties of spaces. In this paper we study group coarse structures (i.e., coarse structures on groups that agree with the algebraic structures), by using group ideals. We introduce a large class of examples of group coarse structures induced by cardinal invariants. In order to enhance the categorical treatment of the subject, we use quasihomomorphisms, as a large-scale counterpart of homomorphisms. In particular, the localisation of a category plays a fundamental role. We then define the notion of functorial coarse structures and we give various examples of those structures.
arXiv (Cornell University), Sep 3, 2015
A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) ... more A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {x ∈ X : vn(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of autochacaracterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proved to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups.
For a subgroup H of a topological abelian group G denote by group S(H) the set of all sequences o... more For a subgroup H of a topological abelian group G denote by group S(H) the set of all sequences of integers (un) such that unh→ 0 for every h ∈ H; H is called t-dense if S(H) = S(G). Motivated by a question of Raczkowski we explore the existence of small (with respect to size or measure) t-dense subgroups of topological abelian groups.
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Papers by Dikran Dikranjan