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。

前辅文
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´edi-Gowers lemma, and the Bourgain-Gamburd expansion machine
   §4.1. The Balog-Szemer´edi-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 SLd(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