Self-similar groups and finite Gelfand pairs
We study the Basilica group \(B\), the iterated monodromy group \(I\) of the complex polynomial \(z^2+i\) and the Hanoi Towers group \(H^{(3)}\). The first two groups act on the binary rooted tree, the third one on the ternary rooted tree. We prove that the action of \(B, I\) and \(H^{(3)}\) on each...
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| Дата: | 2018 |
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| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-8432018-04-04T08:11:51Z Self-similar groups and finite Gelfand pairs D’Angeli, Daniele Donno, Alfredo Rooted \(q\)-ary tree, ultrametric space, fractal group,labelling, rigid vertex stabilizer, 2-points homogeneous action,Gelfand pairs, spherical functions 20E08, 20F65, 20F10, 05C25,43A85, 43A90 We study the Basilica group \(B\), the iterated monodromy group \(I\) of the complex polynomial \(z^2+i\) and the Hanoi Towers group \(H^{(3)}\). The first two groups act on the binary rooted tree, the third one on the ternary rooted tree. We prove that the action of \(B, I\) and \(H^{(3)}\) on each level is 2-points homogeneous with respect to the ultrametric distance. This gives rise to symmetric Gelfand pairs: we then compute the corresponding spherical functions. In the case of \(B\) and \(H^{(3)}\) this result can also be obtained by using the strong property that the rigid stabilizers of the vertices of the first level of the tree act spherically transitively on the respective subtrees. On the other hand, this property does not hold in the case of \(I\). Lugansk National Taras Shevchenko University 2018-03-21 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/843 Algebra and Discrete Mathematics; Vol 6, No 2 (2007) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/843/pdf Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Rooted \(q\)-ary tree ultrametric space fractal group,labelling rigid vertex stabilizer 2-points homogeneous action,Gelfand pairs spherical functions 20E08 20F65 20F10 05C25,43A85 43A90 |
| spellingShingle |
Rooted \(q\)-ary tree ultrametric space fractal group,labelling rigid vertex stabilizer 2-points homogeneous action,Gelfand pairs spherical functions 20E08 20F65 20F10 05C25,43A85 43A90 D’Angeli, Daniele Donno, Alfredo Self-similar groups and finite Gelfand pairs |
| topic_facet |
Rooted \(q\)-ary tree ultrametric space fractal group,labelling rigid vertex stabilizer 2-points homogeneous action,Gelfand pairs spherical functions 20E08 20F65 20F10 05C25,43A85 43A90 |
| format |
Article |
| author |
D’Angeli, Daniele Donno, Alfredo |
| author_facet |
D’Angeli, Daniele Donno, Alfredo |
| author_sort |
D’Angeli, Daniele |
| title |
Self-similar groups and finite Gelfand pairs |
| title_short |
Self-similar groups and finite Gelfand pairs |
| title_full |
Self-similar groups and finite Gelfand pairs |
| title_fullStr |
Self-similar groups and finite Gelfand pairs |
| title_full_unstemmed |
Self-similar groups and finite Gelfand pairs |
| title_sort |
self-similar groups and finite gelfand pairs |
| description |
We study the Basilica group \(B\), the iterated monodromy group \(I\) of the complex polynomial \(z^2+i\) and the Hanoi Towers group \(H^{(3)}\). The first two groups act on the binary rooted tree, the third one on the ternary rooted tree. We prove that the action of \(B, I\) and \(H^{(3)}\) on each level is 2-points homogeneous with respect to the ultrametric distance. This gives rise to symmetric Gelfand pairs: we then compute the corresponding spherical functions. In the case of \(B\) and \(H^{(3)}\) this result can also be obtained by using the strong property that the rigid stabilizers of the vertices of the first level of the tree act spherically transitively on the respective subtrees. On the other hand, this property does not hold in the case of \(I\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/843 |
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AT dangelidaniele selfsimilargroupsandfinitegelfandpairs AT donnoalfredo selfsimilargroupsandfinitegelfandpairs |
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2024-04-12T06:27:31Z |
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2024-04-12T06:27:31Z |
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