Expanders In Group Algebras

Authors:
Roy Meshulam;Avi Wigderson
Affiliations:
Technion, Department of Mathematics, 32000, Haifa, Israel and Institute for Advanced Study, Princeton, USA;Hebrew university, Department of Mathematics, 32000, Jerusalem, Israel and Institute for Advanced Study, Princeton, USA
Venue:
Combinatorica
Year:
2004

Citing 0
Cited 4

A combinatorial construction of almost-ramanujan graphs using the zig-zag product

STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag Product

SIAM Journal on Computing

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Abstract

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.