Expanders In Group Algebras

Let G be a finite group and let p be a prime such that (p, |G|) = 1. We study conditions under which the Abelian group \(\Bbb F\) p [G] has a few G-orbits whose union generate it as an expander (equivalently, all the discrete Fourier coefficients (in absolute value) of this generating set are bounded away uniformly from one).

We prove a (nearly sharp) bound on the distribution of dimensions of irreducible representations of G which implies the existence of such expanding orbits. We further show a class of groups for which such a bound follows from the expansion properties of G. Together, these lead to a new iterative construction of expanding Cayley graphs of nearly constant degree.

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Correspondence to Roy Meshulam.

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Meshulam, R., Wigderson, A. Expanders In Group Algebras. Combinatorica 24, 659–680 (2004). https://doi.org/10.1007/s00493-004-0040-9

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Mathematics Subject Classification (2000):

  • 05C25
  • 20C15