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Expansion in finite simple groups of Lie type / Terence Tao.

By: Material type: TextTextSeries: Graduate studies in mathematics ; v 164.Publication details: Providence : American Mathematical Society, 2015.Description: xi, 303 p. : illustrations (some color) ; 26 cmISBN:
  • 9781470421960 (alk. paper)
Subject(s): DDC classification:
  • 512.482 23 T171
Contents:
1. Expander graphs: Basic theory -- 2. Expansion in Cayley graphs, and Kazhdan's property (T) -- 3. Quasirandom groups -- 4. The Balog-Szemeredi-Gowers lemma, and the Bourgain-Gamburd expansion machine -- 5. Product theorems, pivot arguments, and the Larsen-Pink non-concentration inequality -- 6. Non-concentration in subgroups -- 7. Sieving and expanders -- 8. Related articles -- 8. Cayley graphs the algebra of groups -- 9. The Lang-Weil bound -- 10. The spectral theorem and its converses for unbounded self-adjoint operators 11.Notes on Lie algebras 12. Notes on groups of Lie type -- Bibliography -- Index.
Summary: Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan's property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog-Szemeredi-Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with additional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang-Weil bound, as well as numerous exercises and other optional material.
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Includes bibliographical references and index.

1. Expander graphs: Basic theory --
2. Expansion in Cayley graphs, and Kazhdan's property (T) --
3. Quasirandom groups --
4. The Balog-Szemeredi-Gowers lemma, and the Bourgain-Gamburd expansion machine --
5. Product theorems, pivot arguments, and the Larsen-Pink non-concentration inequality --
6. Non-concentration in subgroups --
7. Sieving and expanders --
8. Related articles --
8. Cayley graphs the algebra of groups --
9. The Lang-Weil bound --
10. The spectral theorem and its converses for unbounded self-adjoint operators
11.Notes on Lie algebras
12. Notes on groups of Lie type --
Bibliography --
Index.

Expander graphs are an important tool in theoretical computer science, geometric group theory, probability, and number theory. Furthermore, the techniques used to rigorously establish the expansion property of a graph draw from such diverse areas of mathematics as representation theory, algebraic geometry, and arithmetic combinatorics. This text focuses on the latter topic in the important case of Cayley graphs on finite groups of Lie type, developing tools such as Kazhdan's property (T), quasirandomness, product estimates, escape from subvarieties, and the Balog-Szemeredi-Gowers lemma. Applications to the affine sieve of Bourgain, Gamburd, and Sarnak are also given. The material is largely self-contained, with additional sections on the general theory of expanders, spectral theory, Lie theory, and the Lang-Weil bound, as well as numerous exercises and other optional material.

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