On conciseness of words in profinite groups

Detomi, Eloisa; Morigi, Marta; Shumyatsky, Pavel

doi:10.1016/j.jpaa.2016.02.003

Cited by 12 publications

(16 citation statements)

References 30 publications

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“…For simplicity and to steer clear of the continuum hypothesis (or Martin's axiom), we record the following proposition that helps us to avoid references to the Baire category theorem, which appear frequently in [2]…”

Section: Preliminariesmentioning

confidence: 99%

“…In [2], Detomi, Morigi and Shumyatsky suggested a strengthened profinite version of Hall's conciseness conjecture, namely that for every word

w

and every profinite group

G

, the bound

false| G_{w} false| ⩽ ℵ_{0}

implies that

w (G)

is finite. They verified this for multilinear commutator words, also known as outer‐commutator words (see Section 3), as well as for the particular words

x^{2}

and

[x^{2}, y]

.…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper, we initiate a systematic investigation of this conjecture and produce positive evidence for it. Among the words treated in [2], the special power word

x^{2}

is the only one for which a simple replacement of the Baire category theorem by Proposition 2.1 yields that it is strongly concise. More work is needed to confirm the Strong Conciseness Conjecture for multilinear commutator words.…”

Section: Introductionmentioning

confidence: 99%

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Strong conciseness in profinite groups

Detomi

Klopsch

Shumyatsky

2020

J. London Math. Soc.

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A group word w is said to be strongly concise in a class C of profinite groups if, for every group G in C such that w takes less than 2 ℵ 0 values in G, the verbal subgroup w(G) is finite. Detomi, Morigi and Shumyatsky established that multilinear commutator words-and the particular words x 2 and [x 2 , y]-have the property that the corresponding verbal subgroup is finite in a profinite group G whenever the word takes at most countably many values in G. They conjectured that, in fact, this should be true for every word. In particular, their conjecture included as open cases power words and Engel words. In the present paper, we take a new approach via parametrised words that leads to stronger results. First we prove that multilinear commutator words are strongly concise in the class of all profinite groups. Then we establish that every group word is strongly concise in the class of nilpotent profinite groups. From this we deduce, for instance, that, if w is one of the group words x 2 , x 3 , x 6 , [x 3 , y] or [x, y, y], then w is strongly concise in the class of all profinite groups. Indeed, the same conclusion can be reached for all words of the infinite families [x m , z1, .

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Section: Preliminariesmentioning

confidence: 99%

“…In [2], Detomi, Morigi and Shumyatsky suggested a strengthened profinite version of Hall's conciseness conjecture, namely that for every word

w

and every profinite group

G

, the bound

false| G_{w} false| ⩽ ℵ_{0}

implies that

w (G)

is finite. They verified this for multilinear commutator words, also known as outer‐commutator words (see Section 3), as well as for the particular words

x^{2}

and

[x^{2}, y]

.…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper, we initiate a systematic investigation of this conjecture and produce positive evidence for it. Among the words treated in [2], the special power word

x^{2}

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Strong conciseness in profinite groups

Detomi

Klopsch

Shumyatsky

2020

J. London Math. Soc.

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“…On the other hand, perhaps the very concept of conciseness in profinite groups can be broadened. It was conjectured in [7] that if w is a word and G a profinite group such that the set G w is countable, then w(G) is finite. The conjecture was confirmed for various words w (see [7] for details) but not for Engel words.…”

Section: Theorem 2 Words Implying Virtual Nilpotency Are Boundedly Cmentioning

confidence: 99%

“…It was conjectured in [7] that if w is a word and G a profinite group such that the set G w is countable, then w(G) is finite. The conjecture was confirmed for various words w (see [7] for details) but not for Engel words. In view of the results obtained in the present paper, the following question seems to be of interest.…”

Section: Theorem 2 Words Implying Virtual Nilpotency Are Boundedly Cmentioning

confidence: 99%

Words of Engel type are concise in residually finite groups

2019

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Given a group-word [Formula: see text] and a group [Formula: see text], the verbal subgroup [Formula: see text] is the one generated by all [Formula: see text]-values in [Formula: see text]. The word [Formula: see text] is said to be concise if [Formula: see text] is finite whenever the set of [Formula: see text]-values in [Formula: see text] is finite. In 1960s, Hall asked whether every word is concise but later Ivanov answered this question in the negative. On the other hand, Hall’s question remains wide open in the class of residually finite groups. In the present paper we show that various generalizations of the Engel word are concise in residually finite groups.

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A note on strong conciseness in virtually nilpotent profinite groups

Detomi

2022

Arch. Math.

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A group word w is said to be strongly concise in a class $${\mathcal {C}}$$ C of profinite groups if, for every group G in $${\mathcal {C}}$$ C such that w takes less than $$2^{\aleph _0}$$ 2 ℵ 0 values in G, the verbal subgroup w(G) is finite. In this paper, we prove that every group word is strongly concise in the class of virtually nilpotent profinite groups.

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On conciseness of words in profinite groups

Cited by 12 publications

References 30 publications

Strong conciseness in profinite groups

Strong conciseness in profinite groups

Words of Engel type are concise in residually finite groups

A note on strong conciseness in virtually nilpotent profinite groups

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