Partition functions of the Ising model on some self-similar Schreier graphs

We study partition functions and thermodynamic limits for the Ising model on three families of finite graphs converging to infinite self-similar graphs. They are provided by three well-known groups realized as automorphism groups of regular rooted trees: the first Grigorchuk's group of intermediate growth; the iterated monodromy group of the complex polynomial $z^2-1$ known as the Basilica group; and the Hanoi Towers group $H^{(3)}$ closely related to the Sierpinsky gasket.