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 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 $I$.