On growth of generalized Grigorchuk's overgroups
Grigorchuk's Overgroup \(\widetilde{\mathcal{G}}\), is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group \(\mathcal{G}\) of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the g...
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Lugansk National Taras Shevchenko University
2020
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oai:ojs.admjournal.luguniv.edu.ua:article-14512021-01-03T08:55:03Z On growth of generalized Grigorchuk's overgroups Samarakoon, S. T. growth of groups, intermediate growth, Grigorchuk group, growth bounds 20E08 Grigorchuk's Overgroup \(\widetilde{\mathcal{G}}\), is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group \(\mathcal{G}\) of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of \(\mathcal{G}\). The group \(\mathcal{G}\), corresponding to the sequence \((012)^\infty = 012012 \dots\), is a member of the family \(\{ G_\omega | \omega \in \Omega = \{ 0, 1, 2 \}^\mathbb{N} \}\) consisting of groups of intermediate growth when sequence \(\omega\) is not eventually constant. Following this construction we define the family \(\{ \widetilde{G}_\omega, \omega \in \Omega \}\) of generalized overgroups. Then \(\widetilde{\mathcal{G}} = \widetilde{G}_{(012)^\infty}\) and \(G_\omega\) is a subgroup of \(\widetilde{G}_\omega\) for each \(\omega \in \Omega\). We prove, if \(\omega\) is eventually constant, then \(\widetilde{G}_\omega\) is of polynomial growth and if \(\omega\) is not eventually constant, then \(\widetilde{G}_\omega\) is of intermediate growth. Lugansk National Taras Shevchenko University 2020-12-30 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1451 10.12958/adm1451 Algebra and Discrete Mathematics; Vol 30, No 1 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1451/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1451/579 Copyright (c) 2020 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2021-01-03T08:55:03Z |
| collection |
OJS |
| language |
English |
| topic |
growth of groups intermediate growth Grigorchuk group growth bounds 20E08 |
| spellingShingle |
growth of groups intermediate growth Grigorchuk group growth bounds 20E08 Samarakoon, S. T. On growth of generalized Grigorchuk's overgroups |
| topic_facet |
growth of groups intermediate growth Grigorchuk group growth bounds 20E08 |
| format |
Article |
| author |
Samarakoon, S. T. |
| author_facet |
Samarakoon, S. T. |
| author_sort |
Samarakoon, S. T. |
| title |
On growth of generalized Grigorchuk's overgroups |
| title_short |
On growth of generalized Grigorchuk's overgroups |
| title_full |
On growth of generalized Grigorchuk's overgroups |
| title_fullStr |
On growth of generalized Grigorchuk's overgroups |
| title_full_unstemmed |
On growth of generalized Grigorchuk's overgroups |
| title_sort |
on growth of generalized grigorchuk's overgroups |
| description |
Grigorchuk's Overgroup \(\widetilde{\mathcal{G}}\), is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group \(\mathcal{G}\) of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of \(\mathcal{G}\). The group \(\mathcal{G}\), corresponding to the sequence \((012)^\infty = 012012 \dots\), is a member of the family \(\{ G_\omega | \omega \in \Omega = \{ 0, 1, 2 \}^\mathbb{N} \}\) consisting of groups of intermediate growth when sequence \(\omega\) is not eventually constant. Following this construction we define the family \(\{ \widetilde{G}_\omega, \omega \in \Omega \}\) of generalized overgroups. Then \(\widetilde{\mathcal{G}} = \widetilde{G}_{(012)^\infty}\) and \(G_\omega\) is a subgroup of \(\widetilde{G}_\omega\) for each \(\omega \in \Omega\). We prove, if \(\omega\) is eventually constant, then \(\widetilde{G}_\omega\) is of polynomial growth and if \(\omega\) is not eventually constant, then \(\widetilde{G}_\omega\) is of intermediate growth. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2020 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1451 |
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AT samarakoonst ongrowthofgeneralizedgrigorchuksovergroups |
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2025-07-17T10:35:10Z |
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2025-07-17T10:35:10Z |
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1837890022887391232 |