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Dikran Dikranjan

Publications and source records attributed to Dikran Dikranjan.

At least 37 records · Page 2Linked to original sources
arXiv2017

Inertial Properties in Groups

Let G be a group and f be an endomorphism of G. A subgroup H of G is called f-inert if the meet of Hf and H has finite index in the image Hf. The subgroups that are f-inert for all inner automorphisms of G are widely known and studied in the literature, under the name inert subgroups. The related notion of inertial endomorphism, namely an endomorphism f such that all subgroups of G are f-inert, was introduced in [30] and thoroughly studied in [31, 33]. The dual notion of fully inert subgroup, namely a subgroup that is f-inert for all endomorphisms of an abelian group A, was introduced in [42] and further studied in [43, 46, 65, 21]. The goal of this paper is to give an overview of up-to-date known results, as well as some new ones, and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra. We survey on classical and recent results on groups whose inner automorphism are inertial. Moreover, we show how inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces, and can be helpful for the computation of the algebraic entropy of continuous endomorphisms.

math.GR↗
arXiv2017

Characterizing Lie groups by controlling their zero-dimensional subgroups

We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The "compact-like" properties we consider include (local) compactness, (local) omega-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is a sample of our characterizations: (i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups. (ii) An abelian topological group G is a Lie group if and only if G is locally minimal, locally precompact and all closed metric zero-dimensional subgroups of G are discrete. (iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups. (iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.

math.GN↗
arXiv2016

Entropy on abelian groups

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.

math.GR↗
arXiv2016

"Varopoulos paradigm": Mackey property vs. metrizability in topological groups

The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: "A metrizable locally convex space carries its Mackey topology ". This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (\cite{AD}, \cite{DMPT}). We say that an abelian group $G$ satisfies the \emph{Varopoulos paradigm} (VP) if any metrizable locally quasi-convex topology on $G$ is the Mackey topology. Surprisingly VP - which is a topological property of the group - is characterizes an algebraic feature, namely being of finite exponent.

math.AT↗
arXiv2015

A complete solution of Markov's problem on connected group topologies

Every proper closed subgroup of a connected Hausdorff group must have index at least c, the cardinality of the continuum. 70 years ago Markov conjectured that a group G can be equipped with a connected Hausdorff group topology provided that every subgroup of G which is closed in all Hausdorff group topologies on G has index at least c. Counter-examples in the non-abelian case were provided 25 years ago by Pestov and Remus, yet the problem whether Markov's Conjecture holds for abelian groups G remained open. We resolve this problem in the positive.

math.GN↗
arXiv2015

Characterized subgroups of topological abelian groups

A subgroup $H$ of a topological abelian group $X$ is said to be characterized by a sequence $\mathbf v =(v_n)$ of characters of $X$ if $H=\{x\in X:v_n(x)\to 0\ \text{in}\ \mathbb T\}$. 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.

math.GN↗
arXiv2015

Direct sums and products in topological groups and vector spaces

We call a subset $A$ of an abelian topological group $G$: (i) $absolutely$ $Cauchy$ $summable$ provided that for every open neighbourhood $U$ of $0$ one can find a finite set $F\subseteq A$ such that the subgroup generated by $A\setminus F$ is contained in $U$; (ii) $absolutely$ $summable$ if, for every family $\{z_a:a\in A\}$ of integer numbers, there exists $g\in G$ such that the net $\left\{\sum_{a\in F} z_a a: F\subseteq A\mbox{ is finite}\right\}$ converges to $g$; (iii) $topologically$ $independent$ provided that $0\not \in A$ and for every neighbourhood $W$ of $0$ there exists a neighbourhood $V$ of $0$ such that, for every finite set $F\subseteq A$ and each set $\{z_a:a\in F\}$ of integers, $\sum_{a\in F}z_aa\in V$ implies that $z_aa\in W$ for all $a\in F$. We prove that: (1) an abelian topological group contains a direct product (direct sum) of $κ$-many non-trivial topological groups if and only if it contains a topologically independent, absolutely (Cauchy) summable subset of cardinality $κ$; (2) a topological vector space contains $\mathbb{R}^{(\mathbb{N})}$ as its subspace if and only if it has an infinite absolutely Cauchy summable set; (3) a topological vector space contains $\mathbb{R}^{\mathbb{N}}$ as its subspace if and only if it has an $\mathbb{R}^{(\mathbb{N})}$ multiplier convergent series of non-zero elements. We answer a question of Hušek and generalize results by Bessaga-Pelczynski-Rolewicz, Dominguez-Tarieladze and Lipecki.

math.GN↗
arXiv2015

On the Borel Complexity of Characterized Subgroups

In a compact abelian group $X$, a characterized subgroup is a subgroup $H$ such that there exists a sequence of characters $\vs=(v_n)$ of $X$ such that $H=\{x\in X:v_n(x)\to 0 \text{ in } \T\}$. Gabriyelyan proved for $X=\T$, that $\{x\in\T:n!x\to 0 \text{ in }\T\}$ is not an $F_σ$-set. In this paper, we give a complete description of the $F_σ$-subgroups of $\T$ characterized by sequences of integers $\vs=(v_n)$ such that $v_n|v_{n+1}$ for all $n\in\N$ (we show that these are exactly the countable characterized subgroups). 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 in the whole group.

math.GN↗
arXiv2014

Final solution of Protasov-Comfort's problem on minimally almost periodic group topologies

We prove that an abelian group admits a minimally almost periodic (MinAP) group topology if and only if it is connected in its Markov-Zariski topology. In particular, every unbounded abelian group admits a MinAP group topology. This answers positively a question set by Comfort, as well as several weaker forms proposed recently by Gabriyelyan. Using this characterization we answer also two open questions of Gould. We prove that a subgroup H of an abelian group G can be realized as the von Neumann kernel of G equipped with some Hausdorff group topology if and only if H is contained in the connected component of zero of G with respect to its Markov-Zariski topology. This completely resolves a question of Gabriyelyan, as well as some of its particular versions which were open.

math.GN↗
arXiv2013

The Bridge Theorem for totally disconnected LCA groups

For a totally disconnected locally compact abelian group, we prove that the topological entropy of a continuous endomorphism coincides with the algebraic entropy of the dual endomorphism with respect to the Pontryagin duality. Moreover, this result is extended to all locally compact abelian groups under the assumption of additivity with respect to some fully invariant subgroups for both the topological and the algebraic entropy.

math.GN↗
arXiv2013

Topological Entropy and Algebraic Entropy for group endomorphisms

The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of locally compact groups, paying special attention to the case of compact and discrete groups respectively. The basic properties of these entropies, as well as many examples, are recalled. Also new entropy functions are proposed, as well as generalizations of several known definitions and results. Furthermore we give some connections with other topics in Mathematics as Mahler measure and Lehmer Problem from Number Theory, and the growth rate of groups and Milnor Problem from Geometric Group Theory. Most of the results are covered by complete proofs or references to appropriate sources.

math.GN↗
arXiv2013

Discrete dynamical systems in group theory

In this expository paper we describe an unifying approach for many known entropies in Mathematics. First we recall the notion of semigroup entropy h_S in the category S of normed semigroups and contractive homomorphisms, recalling also its properties. For a specific category X and a functor F from X to S, we have the entropy h_F, defined by the composition of h_S with F, which automatically satisfies the same properties proved for h_S. This general scheme permits to obtain many of the known entropies as h_F, for appropriately chosen categories X and functors F. In the last part we recall the definition and the fundamental properties of the algebraic entropy for group endomorphisms, noting how its deeper properties depend on the specific setting. Finally we discuss the notion of growth for flows of groups, comparing it with the classical notion of growth for finitely generated groups.

math.GR↗
arXiv2013

Scale function vs Topological entropy

In the realm of topological automorphisms of totally disconnected locally compact groups, the scale function introduced by Willis in \cite{Willis} is compared with the topological entropy. We prove that the logarithm of the scale function is always dominated by the topological entropy and we provide examples showing that this inequality can be strict. Moreover, we give a condition equivalent to the equality between these two invariants. Various properties of the scale function, inspired by those of the topological entropy, are presented.

math.GN↗
arXiv2012

Limit free computation of entropy

Various limit-free formulas are given for the computation of the algebraic and the topological entropy, respectively in the settings of endomorphisms of locally finite discrete groups and of continuous endomorphisms of totally disconnected compact groups. As applications we give new proofs of the connection between the algebraic and the topological entropy in the abelian case and of the connection of the topological entropy with the finite depth for topological automorphisms.

math.DS↗
arXiv2011

Topics in uniform continuity

This paper collects results and open problems concerning several classes of functions that generalize uniform continuity in various ways, including those metric spaces (generalizing Atsuji spaces) where all continuous functions have the property of being close to uniformly continuous.

math.GN↗
arXiv2011

Metrization criteria for compact groups in terms of their dense subgroups

According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup D of a compact abelian group G determines G if the restriction homomorphism G^ --> D^ of the dual groups is a topological isomorphism. We introduce four conditions on D that are necessary for it to determine G and we resolve the following question: If one of these conditions holds for every dense (or G_δ-dense) subgroup D of G, must G be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its G_δ-dense subgroups is metrizable, thereby resolving a question by Hernandez, Macario and Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis CH, the same statement was proved recently by Bruguera, Chasco, Dominguez, Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building G_δ-dense subgroups without uncountable compact subsets in compact groups of weight ω_1 (in ZFC). The construction is delicate, as these subgroups must have non-trivial convergent sequences in some models of ZFC.

math.GN↗
arXiv2011

Arnautov's problems on semitopological isomorphisms

Semitopological isomorphisms of topological groups were introduced by Arnautov, who posed several questions related to compositions of semitopological isomorphisms and the groups G (we call them Arnautov groups) such that for every group topology T on G every semitopological isomorphism with domain (G,T) is necessarily open (i.e., a topological isomorphism). We propose a different approach to these problems by introducing appropriate new notions, necessary for a deeper understanding of Arnautov groups. This allows us to find some partial answers and many examples. In particular, we discuss the relation with minimal groups and non-topologizable groups.

math.GR↗