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We have now seen two ways to construct expander Cayley graphs . The first, discussed in Notes 2, is to use Cayley graphs that are projections of an infinite Cayley graph on a group with Kazhdan’s property (T). The second, discussed in Notes 3, is to combine a quasirandomness property of the group with a flattening hypothesis for the random walk.
We now pursue the second approach more thoroughly. The main difficulty here is to figure out how to ensure flattening of the random walk, as it is then an easy matter to use quasirandomness to show that the random walk becomes mixing soon after it becomes flat. In the case of Selberg’s theorem, we achieved this through an explicit formula for the heat kernel on the hyperbolic plane (which is a proxy for the random walk). However, in most situations such an explicit formula is not available, and one must develop some other tool for forcing flattening, and specifically an estimate of the form
for some , where is the uniform probability measure on the generating set .
In 2006, Bourgain and Gamburd introduced a general method for achieving this goal. The intuition here is that the main obstruction that prevents a random walk from spreading out to become flat over the entire group is if the random walk gets trapped in some proper subgroup of (or perhaps in some coset of such a subgroup), so that remains large for some moderately large . Note that
since , , and is symmetric. By iterating this observation, we seethat if is too large (e.g. of size for some comparable to ), then it is not possible for the random walk to converge to the uniform distribution in time , and so expansion does not occur.
A potentially more general obstruction of this type would be if the random walk gets trapped in (a coset of) an approximate group . Recall that a -approximate group is a subset of a group which is symmetric, contains the identity, and is such that can be covered by at most left-translates (or equivalently, right-translates) of . Such approximate groups were studied extensively in last quarter’s course. A similar argument to the one given previously shows (roughly speaking) that expansion cannot occur if is too large for some coset of an approximate group.
It turns out that this latter observation has a converse: if a measure does not concentrate in cosets of approximate groups, then some flattening occurs. More precisely, one has the following combinatorial lemma:
Lemma 1 (Weighted Balog-Szemerédi-Gowers lemma) Let be a group, let be a finitely supported probability measure on which is symmetric (thus for all ), and let . Then one of the following statements hold:
- (i) (Flattening) One has .
- (ii) (Concentration in an approximate group) There exists an -approximate group in with and an element such that .
This lemma is a variant of the more well-known Balog-Szemerédi-Gowers lemma in additive combinatorics due to Gowers (which roughly speaking corresponds to the case when is the uniform distribution on some set ), which in turn is a polynomially quantitative version of an earlier lemma of Balog and Szemerédi. We will prove it below the fold.
The lemma is particularly useful when the group in question enjoys a product theorem, which roughly speaking says that the only medium-sized approximate subgroups of are trapped inside genuine proper subgroups of (or, contrapositively, medium-sized sets that generate the entire group cannot be approximate groups). The fact that some finite groups (and specifically, the bounded rank finite simple groups of Lie type) enjoy product theorems is a non-trivial fact, and will be discussed in later notes. For now, we simply observe that the presence of the product theorem, together with quasirandomness and a non-concentration hypothesis, can be used to demonstrate expansion:
- (Quasirandomness). The smallest dimension of a nontrivial representation of is at least ;
- (Product theorem). For all there is some such that the following is true. If is a -approximate subgroup with then generates a proper subgroup of ;
- (Non-concentration estimate). There is some even number such that
where the supremum is over all proper subgroups .
Then is a two-sided -expander for some depending only on , and the function (and this constant is in principle computable in terms of these constants).
This criterion for expansion is implicitly contained in this paper of Bourgain and Gamburd, who used it to establish the expansion of various Cayley graphs in for prime . This criterion has since been applied (or modified) to obtain expansion results in many other groups, as will be discussed in later notes.