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)...
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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2007 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2007
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/157371 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Self-similar groups and finite Gelfand pairs / D. D’Angeli, A. Donno // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 2. — С. 54–69. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | 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.
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| ISBN: | 2000 Mathematics Subject Classification: 20E08, 20F65, 20F10, 05C25, 43A85, 43A90. |
| ISSN: | 1726-3255 |