Self-similar groups and finite Gelfand pairs

• Published in: Algebra and Discrete Mathematics
• Date: 2007
• Main Authors: D’Angeli, D., Donno, A.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2007
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/157371
• 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.

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.