The Tits alternative for generalized triangle groups of type \((3,4,2)\)
A generalized triangle group is a group that can be presented in the form \( G = \langle {x,y}\ |{x^p=y^q=w(x,y)^r=1} \rangle \) where \(p,q,r\geq 2\) and \(w(x,y)\) is a cyclically reduced word of length at least \(2\) in the free product \(\mathbb{Z}_p*\mathbb{Z}_q= \langle {x,y}\ |{x^p=y^q=1}\ran...
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| Дата: | 2018 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/826 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| _version_ | 1856543167921258496 |
|---|---|
| author | Howie, James Williams, Gerald |
| author_facet | Howie, James Williams, Gerald |
| author_sort | Howie, James |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-03-22T09:57:42Z |
| description | A generalized triangle group is a group that can be presented in the form \( G = \langle {x,y}\ |{x^p=y^q=w(x,y)^r=1} \rangle \) where \(p,q,r\geq 2\) and \(w(x,y)\) is a cyclically reduced word of length at least \(2\) in the free product \(\mathbb{Z}_p*\mathbb{Z}_q= \langle {x,y}\ |{x^p=y^q=1}\rangle \). Rosenberger has conjectured that every generalized triangle group \(G\) satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple \((p,q,r)\) is one of \((2,3,2),\) \((2,4,2),\) \((2,5,2),\) \((3,3,2),\) \((3,4,2),\) or \((3,5,2)\). Building on a result of Benyash-Krivets and Barkovich from this journal, we show that the Tits alternative holds in the case \((p,q,r)=(3,4,2)\). |
| first_indexed | 2026-02-08T07:58:07Z |
| format | Article |
| id | admjournalluguniveduua-article-826 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:58:07Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-8262018-03-22T09:57:42Z The Tits alternative for generalized triangle groups of type \((3,4,2)\) Howie, James Williams, Gerald Generalized triangle group, Tits alternative, free subgroup 20F05, 20E05, 57M07 A generalized triangle group is a group that can be presented in the form \( G = \langle {x,y}\ |{x^p=y^q=w(x,y)^r=1} \rangle \) where \(p,q,r\geq 2\) and \(w(x,y)\) is a cyclically reduced word of length at least \(2\) in the free product \(\mathbb{Z}_p*\mathbb{Z}_q= \langle {x,y}\ |{x^p=y^q=1}\rangle \). Rosenberger has conjectured that every generalized triangle group \(G\) satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple \((p,q,r)\) is one of \((2,3,2),\) \((2,4,2),\) \((2,5,2),\) \((3,3,2),\) \((3,4,2),\) or \((3,5,2)\). Building on a result of Benyash-Krivets and Barkovich from this journal, we show that the Tits alternative holds in the case \((p,q,r)=(3,4,2)\). Lugansk National Taras Shevchenko University 2018-03-22 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/826 Algebra and Discrete Mathematics; Vol 7, No 4 (2008) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/826/356 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Generalized triangle group Tits alternative free subgroup 20F05 20E05 57M07 Howie, James Williams, Gerald The Tits alternative for generalized triangle groups of type \((3,4,2)\) |
| title | The Tits alternative for generalized triangle groups of type \((3,4,2)\) |
| title_full | The Tits alternative for generalized triangle groups of type \((3,4,2)\) |
| title_fullStr | The Tits alternative for generalized triangle groups of type \((3,4,2)\) |
| title_full_unstemmed | The Tits alternative for generalized triangle groups of type \((3,4,2)\) |
| title_short | The Tits alternative for generalized triangle groups of type \((3,4,2)\) |
| title_sort | tits alternative for generalized triangle groups of type \((3,4,2)\) |
| topic | Generalized triangle group Tits alternative free subgroup 20F05 20E05 57M07 |
| topic_facet | Generalized triangle group Tits alternative free subgroup 20F05 20E05 57M07 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/826 |
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