On the spectrum of self-similar groups and holomorphic dynamics

We will discuss the spectrum of the Laplacian on some self-similar groups, like the Grigorchuk group and the Lamplighter group. Classical Schur renormalization transformations act on spectral parameters as rational maps in two variables. We will show that the spectrum in question can be interpreted as the asymptotic distribution of sliced pullbacks of certain algebraic curves under this rational map. For groups under consideration, this map happens to be fibered over a one-dimensional polynomial. We will discuss an algebra-geometric nature of this integrability phenomenon.
Based upon a joint work with N.-B. Dang and R. Grigorchuk.