內容大鋼
擴展圖是理論電腦科學、幾何群論、概率論和數論中的重要工具。而用於嚴格建立圖的擴展性質的技術來自表示論、代數幾何和算術組合學等數學的不同領域。圍繞后一主題，本書著重討論了Lie型有限群上的Cayley圖的重要情形，發展了諸如Kazhdan性質（T）、擬隨機性、乘積估計、從子簇中逃逸以及Balog-Szemer?di-Gowers引理等工具，還給出了Bourgain、Gamburd和Sarnak的仿射篩法的應用。本書內容在很大程度上是自封的，增加了關於擴展子、譜理論、Lie理論和Lang-Weil界的一般理論的內容，並包含大量習題和其他可選材料。
本書適合對圖論、幾何群論和算術組合學感興趣的研究生和數學研究人員閱讀參考。
目錄
Preface
Notation
Acknowledgments
Part 1.Expansion in Cayley Graphs
Chapter 1.Expander graphs: Basic theory
§1.1.Expander graphs
§1.2.Connection with edge expansion
§1.3.Random walks on expanders
§1.4.Random graphs as expanders
Chapter 2.Expansion in Cayley graphs, and Kazhdan's property (T)
§2.1.Kazhdan's property (T)
§2.2.Induced representations and property (T)
§2.3.The special linear group and property (T)
§2.4.A more elementary approach
Chapter 3.Quasirandom groups
§3.1.Mixing in quasirandom groups
§3.2.An algebraic description of quasirandomness
§3.3.A weak form of Selberg's 3/16 theorem
Chapter 4.The Balog-Szemer?di-Gowers lemma, and the Bourgain-Gamburd expansion machine
§4.1.The Balog-Szemer?di-Gowers lemma
§4.2.The Bourgain-Gamburd expansion machine
Chapter 5.Product theorems, pivot arguments, and the Larsen-Pink nonconcentration inequality
§5.1.The sum-product theorem
§5.2.Finite subgroups of SL2
§5.3.The product theorem in SL2(k)
§5.4.The product theorem in SLa(k)
§5.5.Proof of the Larsen-Pink inequality
Chapter 6.Nonconcentration in subgroups
§6.1.Expansion in thin subgroups
§6.2.Random generators expand
Chapter 7.Sieving and expanders
§7.1.Combinatorial sieving
§7.2.The strong approximation property
§7.3.Sieving in thin groups
Part 2.Related Articles
Chapter 8.Cayley graphs and the algebra of groups
§8.1.A Hall-Witt identity for 2-cocycles
Chapter 9.The Lang-Weil bound
§9.1.The Stepanov-Bombieri proof of the Hasse-Weil bound
§9.2.The proof of the Lang-Weil bound
§9.3.Lang-Weil with parameters
Chapter 10.The spectral theorem and its converses for unbounded self-adjoint operators
§10.1.Self-adjointness and resolvents
§10.2.Self-adjointness and spectral measure
§10.3.Self-adjointness and flows
§10.4.Essential self-adjointness of the Laplace-Beltrami operator
Chapter 11.Notes on Lie algebras
§11.1.Abelian representations
§11.2.Engel's theorem and Lie's theorem
§11.3.Characterising semisimplicity
§11.4.Cartan subalgebras
§11.5.sl2 representations
§11.6.Root spaces
§11.7.Classification of root systems
§11.8.Chevalley bases
§11.9.Casimirs and complete reducibility
Chapter 12.Notes on groups of Lie type
§12.1.Simple Lie groups over C
§12.2.Chevalley groups
§12.3.Finite simple groups of Lie type
Bibliography
Index
[