arXiv subjects

Dikran Dikranjan

Publications and source records attributed to Dikran Dikranjan.

At least 55 records · Page 3Linked to original sources

arXiv2011

Functorial topologies and finite-index subgroups of abelian groups

In the general context of functorial topologies, we prove that in the lattice of all group topologies on an abelian group, the infimum between the Bohr topology and the natural topology is the profinite topology. The profinite topology and its connection to other functorial topologies is the main objective of the paper. We are particularly interested in the poset C(G) of all finite-index subgroups of an abelian group G, since it is a local base for the profinite topology of G. We describe various features of the poset C(G) (its cardinality, its cofinality, etc.) and we characterize the abelian groups G for which C(G)\{G} is cofinal in the poset of all subgroups of G ordered by inclusion. Finally, for pairs of functorial topologies T, S we define the equalizer E(T,S), which permits to describe relevant classes of abelian groups in terms of functorial topologies.

math.GN↗

arXiv2011

Entropy in a category

The Pinsker subgroup of an abelian group with respect to an endomorphism was introduced in the context of algebraic entropy. Motivated by the nice properties and characterizations of the Pinsker subgroup, we generalize its construction in two directions. We introduce the concept of entropy function h of an abelian category and define the Pinsker radical with respect to h, so that the class of all objects with trivial Pinsker radical is the torsion class of a torsion theory.

math.GR↗

arXiv2010

A class of metrizable locally quasi-convex groups which are not Mackey

A topological group $(G,μ)$ from a class $\mathcal G$ of MAP topological abelian groups will be called a {\it Mackey group} in $\mathcal G$ if it has the following property: if $ν$ is a group topology in $G$ such that $(G,ν)\in \mathcal G$ and $(G,ν)$ has the same continuous characters, say $(G,ν)^{\wedge}=(G,μ)^{\wedge}$, then $ν\le μ$. If $\rm{LCS}$ is the class of Hausdorff topological abelian groups which admit a structure of a locally convex topological vector space over $\mathbb R$, it is well-known that every metrizable $(G,μ) \in \rm{LCS}$ is a Mackey group in $\rm{LCS}$. For the class $\rm{LQC}$ of locally quasi-convex Hausdorff topological abelian groups, it was proved in 1999 that every {\bf complete} metrizable $(G,μ)\in \rm{LQC}$ is a Mackey group in $\rm{LQC}$ (\cite{CMPT}). The completeness cannot be \NB dropped within the class $\rm{LQC}$ as we prove in this paper. In fact, we provide a large family of metrizable precompact \NB(noncompact) groups which {\bf are not} Mackey groups in \rm{LQC} (Theorem \ref{basth}). Those examples are constructed from groups of the form $c_0(X)$, whose elements are the null sequences of a topological abelian group $X$, and whose topology is the uniform topology. We first show that for a compact metrizable group $X\ne\{0\}$ the topological group $c_0(X)$ is a non-compact complete metrizable locally quasi-convex group, which has {\bf countable} topological dual iff $X$ is connected. Then we prove that for a connected compact metrizable group $X\ne\{0\}$ the group $c_0(X)$ endowed with the product topology induced from the product $X^{\N}$ is metrizable precompact but not a Mackey group in LQC.

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arXiv2010

A Kronecker-Weyl theorem for subsets of abelian groups

Let N be the set of non-negative integer numbers, T the circle group and c the cardinality of the continuum. Given an abelian group G of size at most 2^c and a countable family F of infinite subsets of G, we construct "Baire many" monomorphisms p: G --> T^c such that p(E) is dense in {y in T^c : ny=0} whenever n in N, E in F, nE={0} and {x in E: mx=g} is finite for all g in G and m such that n=mk for some k in N--{1}. We apply this result to obtain an algebraic description of countable potentially dense subsets of abelian groups, thereby making a significant progress towards a solution of a problem of Markov going back to 1944. A particular case of our result yields a positive answer to a problem of Tkachenko and Yaschenko. Applications to group actions and discrete flows on T^c, diophantine approximation, Bohr topologies and Bohr compactifications are also provided.

math.GR↗

arXiv2010

Strings of group endomorphisms

Recently the strings and the string number of self-maps were used in the computation of the algebraic entropy of special group endomorphisms. We introduce two special kinds of strings, and their relative string numbers. We show that a dichotomy holds for all these three string numbers; in fact, they admit only zero and infinity as values on group endomorphisms.

math.GR↗

arXiv2010

Productivity of sequences with respect to a given weight function

Given a function f: N --> (omega+1)-{0}, we say that a faithfully indexed sequence {a_n: n in N} of elements of a topological group G is: (i) f-Cauchy productive (f-productive) provided that the sequence {prod_{n=0}^m a_n^{z(n)}: m in N} is left Cauchy (converges to some element of G, respectively) for each function z: N --> Z such that |z(n)| <= f(n) for every n in N; (ii) unconditionally f-Cauchy productive (unconditionally f-productive) provided that the sequence {a_{s(n)}: n in N\} is (f\circ s)-Cauchy productive (respectively, (f\circ s)-productive) for every bijection s: N --> N. (Bijections can be replaced by injections here.) We consider the question of existence of (unconditionally) f-productive sequences for a given "weight function" f. We prove that: (1) a Hausdorff group having an f-productive sequence for some f contains a homeomorphic copy of the Cantor set; (2) if a non-discrete group is either locally compact Hausdorff or Weil complete metric, then it contains an unconditionally f-productive sequence for every function f: N--> N; (3) a metric group is NSS if and only if it does not contain an f_omega-Cauchy productive sequence, where f_omega is the function taking the constant value omega. We give an example of an f_omega-productive sequence {a_n: n in N} in a (necessarily non-abelian) separable metric group H with a linear topology and a bijection s: N --> N such that the sequence {prod_{n=0}^m a_{s(n)}: m in N} diverges, thereby answering a question of Dominguez and Tarieladze. Furthermore, we show that H has no unconditionally f_omega-productive sequences. As an application of our results, we resolve negatively a question from C_p(-,G)-theory.

math.GN↗

arXiv2010

Algebraic entropy of shift endomorphisms on abelian groups

For every finite-to-one map $λ:Γ\toΓ$ and for every abelian group $K$, the generalized shift $σ_λ$ of the direct sum $\bigoplus_ΓK$ is the endomorphism defined by $(x_i)_{i\inΓ}\mapsto(x_{λ(i)})_{i\inΓ}$. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of $K$, but mainly on the function $λ$. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.

math.GR↗

arXiv2010

Locally Minimal Topological Groups 2

We continue in this paper the study of locally minimal groups started in \cite{LocMin}. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group $G$ {\it almost minimal} if it has a closed, minimal normal subgroup $N$ such that the quotient group $G/N$ is uniformly free from small subgroups. The class of almost minimal groups includes all locally compact groups, and is contained in the class of locally minimal groups. On the other hand, we provide examples of countable precompact metrizable locally minimal groups which are not almost minimal. Some other significant properties of this new class are obtained.

math.GN↗

arXiv2010

Adjoint algebraic entropy

The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic entropy of the adjoint endomorphism of the Pontryagin dual. As applications, we compute the adjoint algebraic entropy of the shift endomorphisms of direct sums, and we prove an Addition Theorem for the adjoint algebraic entropy of bounded Abelian groups. A dichotomy is established, stating that the adjoint algebraic entropy of any endomorphism can take only values zero or infinity. As a consequence, we obtain the following surprising discontinuity criterion for endomorphisms: every endomorphism of a compact abelian group, having finite positive algebraic entropy, is discontinuous.

math.GR↗

arXiv2010

The Pinsker subgroup of an algebraic flow

The algebraic entropy h, defined for endomorphisms f of abelian groups G, measures the growth of the trajectories of non-empty finite subsets F of G with respect to f. We show that this growth can be either polynomial or exponential. The greatest f-invariant subgroup of G where this growth is polynomial coincides with the greatest f-invariant subgroup P(G,f) of G (named Pinsker subgroup of f) such that h(f|_P(G,f))=0. We obtain also an alternative characterization of P(G,f) from the point of view of the quasi-periodic points of f. This gives the following application in ergodic theory: for every continuous injective endomorphism g of a compact abelian group K there exists a largest g-invariant closed subgroup N of K such that g|_N is ergodic; furthermore, the induced endomorphism g' of the quotient K/N has zero topological entropy.

math.GR↗

arXiv2010

The Markov-Zariski topology of an abelian group

According to Markov, a subset of an abelian group G of the form {x in G: nx=a}, for some integer n and some element a of G, is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (=totally bounded) Hausdorff group topology on G. The family of all algebraic subsets of an abelian group G forms the family of closed subsets of a unique Noetherian T_1 topology on G called the Zariski, or verbal, topology of G. We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Frechet-Urysohn. For a countable family F of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology T on G such that the T-closure of each member of F coincides with its Zariski closure. As an application, we provide a characterization of the subsets of G that are dense in some Hausdorff group topology on G, and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a long-standing problem of Markov.

math.GR↗

arXiv2010

On zero-dimensionality and the connected component of locally pseudocompact groups

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is dense in the component of the completion of G, and G/G_0 is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that G/G_0 may fail to be zero-dimensional even for totally minimal pseudocompact groups.

math.GN↗

arXiv2009

Locally minimal topological groups

A Hausdorff topological group $(G,τ)$ is called locally minimal if there exists a neighborhood $U$ of 0 in $τ$ such that $U$ fails to be a neighborhood of zero in any Hausdorff group topology on $G$ which is strictly coarser than $τ.$ Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudo--convex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. We also prove that a bounded abelian group $G$ admits a non-discrete locally minimal and locally GTG group topology iff $|G|\geq \cont$.

math.GN↗

arXiv2009

Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density

For an uncountable cardinal τand a subset S of an abelian group G, the following conditions are equivalent: (i) |{ns:s\in S}|\ge τfor all integers n\ge 1; (ii) there exists a group homomorphism π:G\to T^{2^τ} such that π(S) is dense in T^{2^τ}. Moreover, if |G|\le 2^{2^τ}, then the following item can be added to this list: (iii) there exists an isomorphism π:G\to G' between G and a subgroup G' of T^{2^τ} such that π(S) is dense in T^{2^τ}. We prove that the following conditions are equivalent for an uncountable subset S of an abelian group G that is either (almost) torsion-free or divisible: (a) S is T-dense in G for some Hausdorff group topology T on G; (b) S is T-dense in some precompact Hausdorff group topology T on G; (c) |{ns:s\in S}|\ge \min{τ:|G|\le 2^{2^τ}} for every integer n\ge 1. This partially resolves a question of Markov going back to 1946.

math.GN↗

arXiv2009

NSS and TAP properties in topological groups close to being compact

We introduce a notion of productivity (summability) of sequences in a topological group G, parametrized by a given function f : N --> omega+1. The extreme case when f is the function taking constant value omega is closely related to the TAP property, the weaker version of the well-known property NSS. We prove that TAP property coincides with NSS in locally compact groups, omega-bounded abelian groups and countably compact minimal abelian groups. As an application of our results, we provide a negative answer to [13, Question 11.1].

math.GN↗

arXiv2009

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f: G --> H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence {σ_n : n\in N} of cardinals such that w(G) = sup {σ_n : n \in N} and sup {2^{σ_n} : n \in N} \leq |G| \leq 2^{w(G)}, where w(G) is the weight of G. If G is an infinite minimal abelian group, then either |G| = 2^σfor some cardinal σ, or w(G) = min {σ: |G| \leq 2^σ}; moreover, the equality |G| = 2^{w(G)} holds whenever cf (w(G)) > ω. For a cardinal κ, we denote by F_κthe free abelian group with κmany generators. If F_κadmits a pseudocompact group topology, then κ\geq c, where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology on F_c is equivalent to the Lusin's Hypothesis 2^{ω_1} = c. For κ> c, we prove that F_κadmits a (zero-dimensional) minimal pseudocompact group topology if and only if F_κhas both a minimal group topology and a pseudocompact group topology. If κ> c, then F_κadmits a connected minimal pseudocompact group topology of weight σif and only if κ= 2^σ. Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology.

math.GN↗

arXiv2008

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that for every locally compact abelian group G, the following statements are equivalent: (i) G contains no sequence {x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G, and x_n --> 0; (ii) one of the subgroups {g \in G : 2g=0 and {g \in G : 3g=0} is open in G; (iii) G contains an open compact subgroup of the form Z_2^κor Z_3^κfor some cardinal κ.

math.GN↗

arXiv2008

Super-sequences in the arc component of a compact connected group

Let G be an abelian topological group. The symbol \hat{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) \subseteq ϕ([-1/4,1/4]) holds only for the trivial character χ\in \hat{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 : \hat{G} --> \hat{G}_a defined by r(χ) = χ\restriction_{G_a} for χ\in \hat{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 \setminus {e} is an infinite suitable set for G in the sense of Hofmann and Morris).

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