arXiv subjects

Dmitri Shakhmatov

Publications and source records attributed to Dmitri Shakhmatov.

At least 19 recordsLinked to original sources

arXiv2024

Coarse structures on locally compact abelian groups

Motivated by the study of the large-scale geometry of topological groups, we investigate particular families of subsets of topological groups named group ideals. We compare different group ideals in the realm of locally compact groups. In particular, we show that a subset of a locally compact abelian group is relatively compact if and only if it is coarsely bounded. Using this result, we prove that an infinite-dimensional Banach space cannot be embedded into any product of locally compact groups.

math.MG↗

arXiv2022

Markov's problem for free groups

We prove that every unconditionally closed subset of a free group is algebraic, thereby answering affirmatively a 76 years old problem of Markov for free groups. In modern terminology, this means that Markov and Zariski topologies coincide in free groups. It follows that the class of groups for which Markov and Zariski topologies coincide is not closed under taking quotients. We also show that Markov and Zariski topologies differ from the so-called precompact Markov topology in non-commutative free groups.

math.GR↗

arXiv2019

Topological groups all continuous automorphisms of which are open

A topological space is reversible if each continuous bijection of it onto itself is open. We introduce an analogue of this notion in the category of topological groups: A topological group G is g-reversible if every continuous automorphism of G (=continuous isomorphism of G onto itself) is open. The class of g-reversible groups contains Polish groups, locally compact sigma-compact groups, minimal groups, abelian groups with the Bohr topology, and reversible topological groups. We prove that subgroups of R^n are g-reversible, for every positive integer n. An example of a compact (so reversible) metric abelian group having a countable dense non-g-reversible subgroup is given. We also highlight the differences between reversible spaces and g-reversible topological groups. Many open problems are scattered throughout the paper.

math.GR↗

arXiv2019

Automorphism groups of dense subgroups of R^n

By an automorphism of a topological group G we mean an isomorphism of G onto itself which is also a homeomorphism. In this article, we study the automorphism group Aut(G) of a dense subgroup G of R^n, n>=1. We show that Aut(G) can be naturally identified with the subgroup I(G)={A in GL(n,R): G A =G} of the group GL(n,R) of all non-degenerated (n x n)-matrices over R, where G A={g A:g in G}. We describe $I(G) for many dense subgroups G of either R or R^2. We consider also an inverse problem of which symmetric subgroups of GL(n,R) can be realized as I(G) for some dense subgroup G of R^n. For example, for n>=2, we show that the group {A in GL(n,R): det A=+-1} cannot be realized in this way. The realization problem is quite non-trivial even in the one-dimensional case and has deep connections to number theory.

math.GR↗

arXiv2019

Cardinal invariants and convergence properties of locally minimal groups

If G is a locally essential subgroup of a compact abelian group K, then: (i) t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K must be metrizable; (iii) G contains a super-sequence S converging to 0 such that |S|=w(G)=w(K). Items (i)--(iii) hold when G is a dense locally minimal subgroup of K. We show that locally minimal, locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007. For every uncountable cardinal kappa, we construct a Frechet-Urysohn minimal group G of character kappa such that the connected component of G is an open normal omega-bounded subgroup (thus, G is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.

math.GN↗

arXiv2019

Compactness properties defined by open-point games

Let S be a topological property of sequences (such as, for example, "to contain a convergent subsequence" or "to have an accumulation point"). We introduce the following open-point game OP(X,S) on a topological space X. In the n'th move, Player A chooses a non-empty open subet U_n of X, and Player B responds by selecting a point x_n in U_n. Player B wins the game if the sequence (x_n) satisfies property S in X; otherwise, Player A wins. The (non-)existence of regular or stationary winning strategies in OP(X,S) for both players defines new compactness properties of the underlying space X. We thoroughly investigate these properties and construct examples distinguishing half of them, for an arbitrary property S sandwiched between sequential compactness and countable compactness.

math.GN↗

arXiv2019

Countably compact groups and sequential order

We use $\diamondsuit$ to construct, for every $α\leqω_1$ a sequential countably compact topological group of sequential order $α$. This establishes the independence of the existence of sequential countably compact non Fréchet groups from the usual axioms of ZFC and answers several questions of D.~Shakhmatov.

math.GN↗

arXiv2019

SSGP topologies on free groups of infinite rank

We prove that every free group G with infinitely many generators admits a Hausdorff group topology T with the following property: for every T-open neighbourhood U of the identity of G, each element g in G can be represented as a product g=g_1 g_2 ... g_k such that the cyclic group generated by each g_i is contained in U. In particular, G admits a Hausdorff group topology with the small subgroup generating property of Gould. This provides a positive answer to a question of Comfort and Gould in the case of free groups with infinitely many generators. The case of free groups with finitely many generators remains open.

math.GN↗

arXiv2018

The impact of the Bohr topology on selective pseudocompactness

Recall that a space X is selectively pseudocompact if for every sequence (U_n) of non-empty open subsets of X one can choose a point x_n in U_n for all n such that the resulting sequence (x_n) has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by García-Ferreira and Ortiz-Castillo, the present name is due to Dorantes-Aldama and the first author. In 2015, García-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact.

math.GN↗

arXiv2018

SSGP topologies on abelian groups of positive finite divisible rank

Let G be an abelian group. For a subset A of G, Cyc(A) denotes the set of all elements x of G such that the cyclic subgroup generated by x is contained in A, and G is said to have the small subgroup generating property (abbreviated to SSGP) if the smallest subgroup of G generated by Cyc(U) is dense in G for every neighbourhood U of zero of G. SSGP groups form a proper subclass of the class of minimally almost periodic groups. Comfort and Gould asked for a characterization of abelian groups G which admit an SSGP group topology, and they solved this problem for bounded torsion groups (which have divisible rank zero). Dikranjan and the first author proved that an abelian group of infinite divisible rank admits an SSGP group topology. In the remaining case of positive finite divisible rank, the same authors found a necessary condition on G in order to admit an SSGP group topology and asked if this condition is also sufficient. We answer this question positively, thereby completing the characterization of abelian groups which admit an SSGP group topology.

math.GN↗

arXiv2017

Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

A space X is selectively sequentially pseudocompact if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in each U_n in such a way that the sequence (x_n) has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of García-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of García-Ferreira and Tomita.

math.GN↗

arXiv2017

The existence of continuous weak selections and orderability-type properties in products and filter spaces

Orderability, weak orderability and the existence of continuous weak selections on filter spaces (i.e., spaces with a single non-isolated point) and their products are discussed. We prove that a closed continuous image X of a suborderable space must be hereditarily paracompact provided that its product X\times Y with some non-discrete space Y has a separately continuous weak selection.

math.GN↗

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↗

arXiv2017

Selectively pseudocompact groups without non-trivial convergent sequences

The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without non-trivial convergent sequences having the following "selective" compactness property: For each free ultrafilter p on N and every sequence {U_n:n in N} of non-empty open subsets of G one can choose a point x_n in U_n for all n in such a way that the resulting sequence {x_n:n in N} has a p-limit in G, that is, {n in N: x_n in V} belongs to p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first author. As a by-product, we show that the free precompact Boolean group over any disjoint sum of maximal countable spaces contains no infinite compact subsets.

math.GN↗

arXiv2017

Selective sequential pseudocompactness

We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a convergent subsequence. We show that the class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by García-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every omega-bounded (=the closure of which countable set is compact) group is SSP, while compact spaces need not be SSP. Finally, we construct SSP group topologies on both the free group and the free Abelian group with continuum-many generators.

math.GN↗

arXiv2016

Completeness and compactness properties in metric spaces, topological groups and function spaces

We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to function spaces C_p(X,G) of G-valued continuous functions on a space X with the topology of pointwise convergence, for a separable metric group G. Not only the results but also the proofs themselves are novel even in the classical case when G is the real line. A space X is weakly pseudocompact if it is G_delta-dense in at least one of its compactifications. A topological group G is precompact if it is topologically isomorphic to a subgroup of a compact group. We prove that every weakly pseudocompact precompact topological group is pseudocompact, thereby answering positively a question of Tkachenko.

math.GN↗

arXiv2015

A countable free closed non-reflexive subgroup of Z^c

We prove that the group G=Hom(P,Z) of all homomorphisms from the Baer-Specker group P to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G is discrete. As G is non-discrete, it is not reflexive. Since G can be viewed as a closed subgroup of the Tychonoff product of continuum many copies of the integers Z, this provides an example of a group described in the title, thereby answering Problem 11 from [J.Galindo, L.Recorder-Núñez, M.Tkachenko, Reflexivity of prodiscrete topological groups, J. Math. Anal. Appl. 384 (2011), 320--330.] It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.

math.GN↗

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↗