扩展图是理论计算机科学、几何群论、概率论和数论中的重要工具。而用于严格建立图的扩展性质的技术来自表示论、代数几何和算术组合学等数学的不同领域。围绕后一主题，本书着重讨论了Lie型有限群上的Cayley图的重要情形，发展了诸如Kazhdan性质（T）、拟随机性、乘积估计、从子簇中逃逸以及Balog-Szemerédi-Gowers引理等工具，还给出了Bourgain、Gamburd和Sarnak的仿射筛法的应用。本书内容在很大程度上是自封的，增加了关于扩展子、谱理论、Lie理论和Lang-Weil界的一般理论的内容，并包含大量习题和其他可选材料。
本书适合对图论、几何群论和算术组合学感兴趣的研究生和数学研究人员阅读参考。

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-Szemerédi-Gowers lemma, and the Bourgain-Gamburd expansion machine
§4.1.The Balog-Szemerédi-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