The Thompson group F is one of the most famous and most important groups related to many areas of mathematics. The open question about amenability of this group can be expressed in terms of spectra of Markov operators of random walks on F. Moreover, amenability of F would imply amenability of all Schreier graphs of F. This motivates studying spectral properties of these graphs.

In this talk I will present results on spectral properties of the family of Schreier graphs associated to the action of the Thompson group F on the interval [0,1]. In particular, I will describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to them. As a byproduct we will obtain the asymptotics of the return probabilities of the simple random walk on the orbit of 1/2. The talk is based on a joint work with Rostislav Grigorchuk.