Abstract
Let the group G act transitively on the finite set \(\Omega \), and let \(S \subseteq G\) be closed under taking inverses. The Schreier graph \(Sch(G \circlearrowleft \Omega ,S)\) is the graph with vertex set \(\Omega \) and edge set \(\{ (\omega ,\omega ^s) : \omega \in \Omega , s \in S \}\). In this paper, we show that random Schreier graphs on \(C \log |\Omega |\) elements exhibit a (two-sided) spectral gap with high probability, magnifying a well-known theorem of Alon and Roichman for Cayley graphs. On the other hand, depending on the particular action of G on \(\Omega \), we give a lower bound on the number of elements which are necessary to provide a spectral gap. We use this method to estimate the spectral gap when G is nilpotent.
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Notes
We work in the general context of multisets, which is useful in random arguments. A multiset is symmetric if each element appears with the same multiplicity as its inverse. Moreover, we write \(S^{-1}:=\{ s^{-1} : s \in S \}\).
Unless explicitly stated otherwise, all logarithms are natural.
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Acknowledgements
The author thanks Pablo Spiga and the two anonymous referees for many useful comments and remarks.
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The Funding was provided by Universitá degli Studi di Firenze, Istituto Nazionale di Alta Matematica "Francesco Severi".
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Sabatini, L. Random Schreier graphs and expanders. J Algebr Comb 56, 889–901 (2022). https://doi.org/10.1007/s10801-022-01136-z
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DOI: https://doi.org/10.1007/s10801-022-01136-z