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

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Artem Dudko and 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 $L$.## References

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## Additional Information

**Artem Dudko**- Affiliation: IMPAN, Sniadeckich 8, 00-656 Warsaw, Poland
- MR Author ID: 790914
- ORCID: 0000-0003-2706-3215
- Email: adudko@impan.pl
**Rostislav Grigorchuk**- Affiliation: Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 193739
- ORCID: 0000-0002-3067-5477
- Email: grigorch@math.tamu.edu
- Received by editor(s): December 6, 2021
- Received by editor(s) in revised form: August 11, 2022
- Published electronically: January 24, 2023
- Additional Notes: The first author is the corresponding author

The second author was partially supported by Simons Foundation Collaboration Grant for Mathematicians, Award Number 527814. Also, the second author was supported by the Max Planck Institute for Mathematics in Bonn and Humboldt Foundation - © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 2787-2819 - MSC (2020): Primary 20F65, 05C50
- DOI: https://doi.org/10.1090/tran/8806
- MathSciNet review: 4557881