Varieties of groups and the problem on conciseness of words
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911855870476288 |
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| author | Acciarri, Cristina Shumyatsky, Pavel |
| author_facet | Acciarri, Cristina Shumyatsky, Pavel |
| contents | A group-word $w$ is concise in a class of groups $\mathcal X$ if and only if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G\in \mathcal X$. It is a long-standing open problem whether every word is concise in residually finite groups. In this paper we observe that the conciseness of a word $w$ in residually finite groups is equivalent to that in the class of virtually pro-$p$ groups.
This is used to show that if $q,n$ are positive integers and $w$ is a multilinear commutator word, then the words $w^q$ and $[w^q,_{n} y]$ are concise in residually finite groups. Earlier this was known only in the case where $q$ is a prime power.
In the course of the proof we establish that certain classes of groups satisfying the law $w^q\equiv1$, or $[δ_k^q,{}_n\, y]\equiv1$, are varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02209 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Varieties of groups and the problem on conciseness of words Acciarri, Cristina Shumyatsky, Pavel Group Theory 20E26, 20F10, 20F40, 20E10, 20F45 A group-word $w$ is concise in a class of groups $\mathcal X$ if and only if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G\in \mathcal X$. It is a long-standing open problem whether every word is concise in residually finite groups. In this paper we observe that the conciseness of a word $w$ in residually finite groups is equivalent to that in the class of virtually pro-$p$ groups. This is used to show that if $q,n$ are positive integers and $w$ is a multilinear commutator word, then the words $w^q$ and $[w^q,_{n} y]$ are concise in residually finite groups. Earlier this was known only in the case where $q$ is a prime power. In the course of the proof we establish that certain classes of groups satisfying the law $w^q\equiv1$, or $[δ_k^q,{}_n\, y]\equiv1$, are varieties. |
| title | Varieties of groups and the problem on conciseness of words |
| topic | Group Theory 20E26, 20F10, 20F40, 20E10, 20F45 |
| url | https://arxiv.org/abs/2308.02209 |