On spectral properties of the Schreier graphs of the Thompson group $F$

Artem Dudko; Rostislav Grigorchuk

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arXiv · 2107.01109

On spectral properties of the Schreier graphs of the Thompson group $F$

Artem Dudko·Rostislav Grigorchuk

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Abstract

In this article we study spectral properties of the family of Schreier graphs associated to the action of the Thompson group $F$ on the interval [0,1]. In particular, we describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to the Schreier graph $\Upsilon$ of the orbit of 1/2. As a byproduct we calculate the asymptotics of the return probabilities of the simple random walk on $\Upsilon$ starting at 1/2. In addition, given a Laplace type operator $L$ on a tree-like graph we study relations between the spectral measures of $L$ associated to delta functions of different vertices and the spectrum of $L$.

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BibTeX RIS

Artem Dudko, Rostislav Grigorchuk. 2021-07-02. On spectral properties of the Schreier graphs of the Thompson group $F$. https://arxiv.org/abs/2107.01109

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