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• W2182393699 abstract "This expository paper provides an overview of expanders and the various forms of the expan- sion property: graph edge expansion, spectral gap, rapid mixing, and connections to Property (T ) and Selberg's 3 16 Theorem. Some key ideas are also discussed from recent results on expan- sion in linear algebraic groups. rst two known explicit constructions relied on deep results in representation theory and in the spectral theory of modular forms, respectively. Originally considered in computer science in the context of networks, expanders have found surpris- ing connections to and applications in various branches of pure mathematics; see (22) and references therein. This expository paper aims to present expanders in various forms and to discuss recent related results. The paper is organized as follows. Section 2 introduces expanders and looks at several equivalent reformulations of the expander property. Section 3 takes up the explicit constructions expanders using Property (T ), Property ( ), and Selberg's 3 16 Theorem. In particular, we will see that for every n 2, Cayley graphs of SLn(Z=pZ) with respect to projections of certain xed generators of SLn(Z) form families of expanders. Much of the material in these sections is based on Lubotzky's exposition (22), omitting some technical proofs. Section 4 looks at recent results on expansion in certain linear algebraic groups, initiated by Helfgott's work in 2005. We discuss Sarnak and Xue's idea for exhibiting a spectral gap, as well as key components of Bourgain and Gamburd's characterization of expanders for certain Cayley graphs of SL2(Z=pZ), and several generalizations of this result. Finally, Section 5 discusses the diameter of expanders and a related open problem." @default.
• W2182393699 created "2016-06-24" @default.
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• W2182393699 date "2011-01-01" @default.
• W2182393699 modified "2023-09-27" @default.
• W2182393699 title "Expanders and Related Results" @default.
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