Markov's Problems Through the Looking Glass of Zariski and Markov Topologies

Dikranjan, Dikran; Toller, Daniele

doi:10.1142/9789814350051_0008

Cited by 8 publications

(19 citation statements)

References 39 publications

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“…In this paper, we answer [7, Question 39], repeated also as [11,Question 8.1], by confirming Markov's conjecture for all free (non-commutative) groups.…”

Section: Introductionsupporting

confidence: 74%

“…Let G be a non-commutative free group. Then the space (G, Z G ) is Noetherian; that is, each strictly decreasing sequence of its closed subsets stabilises [11,Corollary 5.2]. Therefore, all Hausdorff subspaces of (G, Z G ) are finite.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…Dikranjan and Toller in [11,Section 5.3] derived the following explicit description of the Zariski topology of free groups from results of Bryant [4], Guba [15], and Chiswell and Remeslennikov [5]: It was noted in [25,Section 5.4.1] that, without loss of generality, one can take c = 1 in Theorem 1.1. We also note that the set C G (b) in Theorem 1.1 is an infinite cyclic subgroup of G containing b but it need not coincide with the cyclic subgroup of G generated by b.…”

Section: Introductionmentioning

confidence: 99%

“…Its Corollary 8.7 states that if a group G is maximally almost periodic group in its discrete topology, then B G and P G induce the same (subspace) topology on every Bohr compact subset of G. Another Corollary 8.8 combines Theorem 8.6 with the result of Thom [24] on the existence of a non-trivial sequence in the free group with two generators converging to its identity element in the Bohr topology to conclude that a non-commutative free group G has an infinite Hausdorff subspace in its precompact Markov topology P G . On the other hand, the Zariski topology Z G of a free group G is known to be Noetherian [11,Corollary 5.2], and Noetherian spaces have no infinite Hausdorff subspaces. It follows that Z G = P G for a non-commutative free group G. Combining this with our Theorem 1.2, we deduce Theorem 1.3 in Section 9.…”

Section: Introductionmentioning

confidence: 99%

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Markov's problem for free groups

Shakhmatov¹,

Yáñez²

2021

Preprint

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“…In this paper, we answer [7, Question 39], repeated also as [11,Question 8.1], by confirming Markov's conjecture for all free (non-commutative) groups.…”

Section: Introductionsupporting

confidence: 74%

Section: Proof Of Theorem 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Markov's problem for free groups

Shakhmatov¹,

Yáñez²

2021

Preprint

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show abstract

“…The Zariski topology was explicitly introduced only in 1977 by Bryant [5] under the name verbal topology; the term was introduced by Dikranjan and Shakhmatov [6]. This topology was studied in, e.g., [5,25,8,10,11].…”

Section: Problem 12 Find Necessary and Sufficient Conditions For Thmentioning

confidence: 99%

Topologizations of a set endowed with an action of a monoid

Банах

Протасов

Sipacheva

2014

Topology and its Applications

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Abstract. Given a set X and a family G of self-maps of X, we study the problem of the existence of a nondiscrete Hausdorff topology on X with respect to which all functions f ∈ G are continuous. A topology on X with this property is called a G-topology. The answer is given in terms of the Zariski G-topology ζ G on X, that is, the topology generated by the subbase consisting of the sets {x ∈ X : f (x) = g(x)} and {x ∈ X : f (x) = c}, where f, g ∈ G and c ∈ X. We prove that, for a countable monoid G ⊂ X X , X admits a non-discrete Hausdorff G-topology if and only if the Zariski G-topology ζ G is non-discrete; moreover, in this case, X admits 2 c hereditarily normal G-topologies.

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Order and minimality of some topological groups

Megrelishvili

Polev

2016

Topology and its Applications

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We study the minimality of some natural matrix groups defined on nice topological fields F. More precisely, we examine the minimality of the special upper triangular groups SUT(n, F), the special linear groups SL(n, F) and the projective general linear groups PGL(n, F). We prove that if F is a local field of characteristic different than 2, then SUT(n, F) is minimal for every n ∈ N. This result is new even for F = R and n = 2. In contrast, we show that SUT(3, Q(i)) is not minimal, where Q(i) is the Gaussian rational field.Using Iwasawa decomposition our technique leads to the total minimality of SL(n, F) for local fields F of characteristic different than 2. Note that recent results of Bader and Gelander [2], obtained in a different way, imply that the same is true for every characteristic. This extends the well known result of Remus and Stoyanov [19] about the total minimality of SL(n, R).Moreover, we show that if F is a local field, then PGL(n, F) is totally minimal for every n ∈ N.

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Markov's Problems Through the Looking Glass of Zariski and Markov Topologies

Cited by 8 publications

References 39 publications

Markov's problem for free groups

Markov's problem for free groups

Topologizations of a set endowed with an action of a monoid

Order and minimality of some topological groups

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