Proceedings of Symposia in Pure Mathematics Volume 76.2, 2007 A New Approach to Spectral Gap Problems Jean Bourgain To Barry ABSTRACT. Based on purely analytical methods, we exhibit new families of expanders in SL2(p) (p prime) and SU(2), contributing to conjectures of Lubotzky and Sarnak. This is a report on joint work with Gamburd. CONTENTS 1. Introduction 2. Results 3. Methods 4. Further Remarks References 1. Introduction Given an undirected d-regular graph Q and a subset X of V, the expansion of X,c(X), is defined to be the ratio |9(X)|\|X|, where d(X) — {y G Q: distance (y, X) = 1}. The expansion coefficient of a graph Q is defined as follows: c(G)=mi{c(X)\ \X\±\g\}. A family of d-regular graphs Qn^ forms a family of C-expanders if there is a fixed positive constant C, such that liminf c(gn,d)C. (1.1) n—-oo The adjacency matrix of Q, A(Q), is the \Q\ x \Q\ matrix, with rows and columns indexed by vertices of 5, such that the x, y entry is 1 if and only if x and y are ad- jacent and 0 otherwise. Using the discrete Cheeger-Buser inequality, the condition 2000 Mathematics Subject Classification. 22E45, 42Axx, 54H15, 81Q30. Key words and phrases. Hecke operators, sphere,rotations, spectrum. The author was supported in part by NSF grant DMS-0401277 and a Sloan Foundation Fellowship. ©2007 American Mathematical Society 499 http://dx.doi.org/10.1090/pspum/076.2/2307745

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