  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.
