Diameters, distortion and eigenvalues

We study the relation between the diameter, the first positive eigenvalue of the discrete $p$-Laplacian and the $\ell_p$-distortion of a finite graph. We prove an inequality relating these three quantities and apply it to families of Cayley and Schreier graphs. We also show that the $\ell_p$-distortion of Pascal graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain estimates for the convergence to zero of the spectral gap as an application of the main result.Comment: Final version, to appear in the European Journal of Combinatoric

Asymptotic aspects of Schreier graphs and Hanoi Towers groups

We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior.Comment: 5 page

On the Spectrum of Hecke Type Operators related to some Fractal Groups

We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common ``finite approximation'' method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also ``substitutional graphs''. We also formulate our results in terms of Hecke type operators related to some irreducible quasi-regular representations of fractal groups and in terms of the Markovian operator associated to noncommutative dynamical systems via which these fractal groups were originally defined. In the computations we performed, the self-similarity of the groups is reflected in the self-similarity of some operators; they are approximated by finite counterparts whose spectrum is computed by an ad hoc factorization process.Comment: 1 color figure, 2 color diagrams, many figure

From self-similar groups to self-similar sets and spectra

The survey presents developments in the theory of self-similar groups leading to applications to the study of fractal sets and graphs, and their associated spectra