陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「254B, Notes 2: Cayley graphs and Kazhdan’s property (T)」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

In the previous set of notes we introduced the notion of expansion in arbitrary -regular graphs. For the rest of the course, 下面会 now focus attention primarily to a special type of -regular graph, namely a Cayley graph .

Definition 1 (Cayley graph) Let be a group, and let be a finite subset of . We assume that is symmetric (thus whenever ) and does not contain the identity (this is to avoid loops). Then the (right-invariant) Cayley graph is defined to be the graph with vertex set and edge set , thus each vertex is connected to the elements for , and so is a -regular graph.

已知结果和反例

Example 2 The graph in Exercise 3 of Notes 1 is the Cayley graph on with generators .

Remark 3 We call the above Cayley graphs right-invariant because every right translation on is a graph automorphism of . This group of automorphisms acts transitively on the vertex set of the Cayley graph. One can thus view a Cayley graph as a homogeneous space of , as it “looks the same” from every vertex. One could of course also consider left-invariant Cayley graphs , in which is connected to rather than . However, the two such graphs are isomorphic using the inverse map ,

证明或构造的主线

Remark 4 For minor technical reasons, it will be convenient later on to allow to contain the identity and to come with multiplicity (i.e. it will be a multiset rather than a set). If one does so, of course, the resulting Cayley graph will now contain some loops and multiple edges. For the purposes of building expander families, we would of course want the underlying group to be finite. However, it will be convenient at various times to “lift” a finite Cayley graph up to an in

We will also sometimes consider a generalisation of a Cayley graph, known as a Schreier graph :

阅读时建议盯住的点

Definition 5 (Schreier graph) Let be a finite group that acts (on the left) on a space , thus there is a map from to such that and for all and . Let be a symmetric subset of which acts freely on in the sense that for all and , and for all distinct and . Then the Schreier graph is defined to be the graph with vertex set and edge set .

Example 6 Every Cayley graph is also a Schreier graph , using the obvious left-action of on itself. The -regular graphs formed from permutations that were studied in the previous set of notes is also a Schreier graph provided that for all distinct , with the underlying group being the permutation group (which acts on the vertex set in the obvious manner), and .

值得单独记下的条目

• (i) Show that is a one-sided -expander for for some if and only if generates .
• (ii) Show that is a two-sided -expander for for some if and only if generates , and furthermore intersects each index subgroup of .
• Show that the are a two-sided expander family if and only if there exists a such that for all sufficiently large , one has for some , where denotes the convolution of copies of .
• Show that the are a one-sided expander family if and only if there exists a such that for all sufficiently large , one has for some .
• (iv) One has and for all .
• (v) If generates as a group (thus every element of is a finite word in ), show that has Kazhdan property (T) if and only if .
• (ii) There exists a unitary representation with no non-trivial invariant vectors such that .
• (ii) For any unitary representation (possibly containing invariant vectors), and any , one has

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源