On profinite groups admitting a word with only few values
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911782586548224 |
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| author | Shumyatsky, Pavel |
| author_facet | Shumyatsky, Pavel |
| contents | A group-word $w$ is called concise if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G$. It is known that there are words that are not concise. The problem whether every word is concise in the class of profinite groups remains wide open. Moreover, there is a conjecture that every word $w$ is strongly concise in profinite groups, that is, $w(G)$ is finite whenever $G$ is a profinite group in which $w$ takes less than $2^{\aleph_0}$ values. In this paper we show that if the word $w$ takes less than $2^{\aleph_0}$ values in a profinite group $G$ then $w(w(G))$ is finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15707 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On profinite groups admitting a word with only few values Shumyatsky, Pavel Group Theory A group-word $w$ is called concise if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G$. It is known that there are words that are not concise. The problem whether every word is concise in the class of profinite groups remains wide open. Moreover, there is a conjecture that every word $w$ is strongly concise in profinite groups, that is, $w(G)$ is finite whenever $G$ is a profinite group in which $w$ takes less than $2^{\aleph_0}$ values. In this paper we show that if the word $w$ takes less than $2^{\aleph_0}$ values in a profinite group $G$ then $w(w(G))$ is finite. |
| title | On profinite groups admitting a word with only few values |
| topic | Group Theory |
| url | https://arxiv.org/abs/2401.15707 |