Self-similar groups and holomorphic dynamics: Renormalization, integrability, and spectrum

TL;DR

This paper investigates the spectral measures of Laplacians on Schreier graphs of self-similar groups using dynamical systems and algebraic geometry, revealing integrability and spectral distribution properties.

Contribution

It introduces a novel approach linking spectral measures to algebraic curves and rational maps, uncovering integrability phenomena in self-similar group actions.

Findings

Spectral measures relate to iterated pullbacks of algebraic curves.

Dynamical criteria for spectrum discreteness are established.

Explicit convergence rates for spectral approximations are provided.

Abstract

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current). We follow up with a dynamical criterion for discreteness of the spectrum. In case of discrete spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.